integration

The Fundamental Theorem of Calculus: How Differentiation and Integration Become Two Sides of the Same Idea

August 12, 2026 by Splendid Leave a Comment

One of the most remarkable discoveries in mathematics is that differentiation and integration, which initially appear to be completely different operations, are deeply connected.

Differentiation tells us how something is changing.

Integration tells us how small changes accumulate.

The Fundamental Theorem of Calculus (FTC) reveals that these two ideas are essentially inverse processes.

This theorem is the bridge that turns integration from an abstract idea about accumulation and area into a practical computational tool.

1. Two seemingly different questions

Suppose we have a function representing the rate at which something changes.

For example, imagine that a car’s velocity is

$v(t)=2t$

measured in metres per second.

We might ask:

How quickly is the car moving at a particular instant?

That is a differentiation-type question when we start with position and find velocity.

But we might instead ask:

How far has the car travelled during a particular period?

Now we need to accumulate velocity over time.

That is an integration problem.

So we have two fundamentally different-looking operations:

$\text{Differentiation}\rightarrow\text{rate of change}$ $\text{Integration}\rightarrow\text{accumulation}$

The remarkable discovery is that these operations undo each other.

2. Differentiation starts with accumulation and finds the rate

Imagine that we know the total distance travelled by a car at every instant.

Call the distance function $s(t)$ .

The velocity is the rate at which distance changes:

$v(t)=s'(t)$

So differentiation takes us from:

$\text{total accumulated quantity}\rightarrow\text{rate of change}$

For example, if

$s(t)=t^2$

then

$s'(t)=2t$

Therefore the velocity is

$v(t)=2t$

3. Integration goes in the opposite direction

Now suppose we know velocity instead:

$v(t)=2t$

How can we recover the distance travelled?

We integrate:

$s(t)=\int 2t,dt$

Using the power rule for integration,

$\int 2t,dt=t^2+C$

So:

$s(t)=t^2+C$

Differentiating this gives us back:

$s'(t)=2t$

The constant $C$ disappears during differentiation.

This is our first glimpse of the relationship:

$\frac{d}{dx}\left(\int f(x),dx\right)=f(x)$

In simple terms:

Differentiate what you integrate, and you get back the original function.

But the Fundamental Theorem goes much deeper.

4. Integration is really accumulation

Consider a function $f(x)$ .

Suppose we want to know how much of the quantity represented by $f(x)$ has accumulated from $a$ to $x$ .

Define a new function:

$F(x)=\int_a^x f(t),dt$

This says:

Start at $a$ and accumulate the values of $f(t)$ until we reach $x$ .

Geometrically, if $f(x)$ is positive, this represents the area under the curve between $a$ and $x$ .

Now comes the astonishing part.

Differentiate this accumulated area:

$F'(x)=f(x)$

The rate at which accumulated area is increasing is exactly the height of the function at that point.

That is the heart of the Fundamental Theorem of Calculus.

5. Why does this make intuitive sense?

Suppose we have accumulated area up to $x$ .

Now increase $x$ by a very small amount $\Delta x$ .

We add a thin strip of area.

Its approximate area is:

$\Delta A\approx f(x)\Delta x$

Therefore:

$\frac{\Delta A}{\Delta x}\approx f(x)$

As $\Delta x$ becomes smaller and smaller:

$\frac{dA}{dx}=f(x)$

So the derivative of accumulated area is simply the height of the curve.

This is not a coincidence.

It is exactly why differentiation and integration are connected.

6. The first Fundamental Theorem

$F(x)=\int_a^x f(t),dt$

then, under the usual continuity conditions,

$F'(x)=f(x)$

This means that integration can actually construct a function whose derivative is the original function.

If we know $f(x)$ , we can construct:

$F(x)=\int_a^x f(t),dt$

and then:

$F'(x)=f(x)$

This gives a powerful interpretation of integration:

Integration is a process for constructing an antiderivative.

7. But what about the mysterious +C?

When we calculate an indefinite integral, we write:

$\int f(x),dx=F(x)+C$

Why?

Because differentiation cannot detect constants.

For example:

$\frac{d}{dx}(x^2)=2x$

But also:

$\frac{d}{dx}(x^2+5)=2x$

and:

$\frac{d}{dx}(x^2-100)=2x$

Every function of the form

$x^2+C$

has the same derivative.

Therefore:

$\int 2x,dx=x^2+C$

The constant represents the fact that differentiation loses information.

8. Definite integration changes the situation

Now consider:

$\int_a^b f(x),dx$

This is a definite integral.

Unlike an indefinite integral, it produces a number rather than a family of functions.

For example:

$\int_0^3 2x,dx$

We know that an antiderivative of $2x$ is:

$x^2$

The Fundamental Theorem tells us that:

$\int_a^b f(x),dx=F(b)-F(a)$

where $F'(x)=f(x)$ .

Therefore:

$\int_0^3 2x,dx=3^2-0^2=9$

We have converted an accumulation problem into two evaluations of an antiderivative.

That is an enormous simplification.

9. Why this theorem is so powerful

Imagine trying to calculate the area under a curve by adding infinitely many tiny rectangles.

That is conceptually what integration does.

We could write:

$\text{Area}\approx\sum f(x_i)\Delta x$

and then make the rectangles thinner and thinner.

In the limit:

$\text{Area}=\int_a^b f(x),dx$

This seems to suggest that we need to perform an infinite limiting process every time we want an area.

But the Fundamental Theorem gives us a shortcut.

Find an antiderivative:

$F'(x)=f(x)$

Then simply calculate:

$\int_a^b f(x),dx=F(b)-F(a)$

So instead of explicitly adding infinitely many tiny pieces, we can find a function whose derivative is the original function.

This is one of the great computational ideas in mathematics.

10. The theorem connects two worlds

The Fundamental Theorem connects:

Local behaviour

with

Global accumulation

A derivative describes what is happening locally.

An integral describes what has accumulated globally.

Yet the theorem tells us:

$\text{local rate of change}\leftrightarrow\text{global accumulation}$

This is one reason calculus is so powerful.

A tiny local relationship can tell us something about an entire interval.

11. A beautiful example: velocity and distance

Suppose a particle has velocity:

$v(t)=3t^2$

We want to know how far it travels between $t=1$ and $t=4$ .

Distance is accumulated velocity:

$\text{Distance}=\int_1^4 3t^2,dt$

An antiderivative is:

$t^3$

Therefore:

$\int_1^4 3t^2,dt=4^3-1^3=63$

So the particle travels 63 units.

Notice what happened.

We did not need to calculate the position function first.

We simply accumulated the velocity.

This is the Fundamental Theorem in action.

12. Another example: changing revenue

Suppose a business has a marginal revenue function:

$MR(q)=100-2q$

Marginal revenue tells us approximately how much additional revenue is generated by selling one additional unit.

Suppose we want the change in revenue when quantity increases from $q=10$ to $q=30$ .

We accumulate marginal revenue:

$\Delta R=\int_{10}^{30}(100-2q),dq$

An antiderivative is:

$100q-q^2$

Therefore:

$\Delta R=[100q-q^2]_{10}^{30}$

So:

$\Delta R=(3000-900)-(1000-100)=1200$

The business’s revenue increases by 1200 monetary units over that quantity range.

This illustrates why integration is so important in economics.

A marginal quantity tells us a rate of change.

Integration reconstructs the total change.

13. Integration and differentiation are not merely opposite tricks

It is tempting to say:

“Integration is just differentiation backwards.”

That is useful, but incomplete.

The deeper idea is:

$\text{Differentiation measures change}$

while:

$\text{Integration measures accumulation}$

The Fundamental Theorem says that these two perspectives are mathematically connected.

If something accumulates according to a rate $f(x)$ , then the derivative of that accumulated quantity is $f(x)$ .

And if $F(x)$ has derivative $f(x)$ , then the accumulated change in $F$ over an interval is obtained by integrating $f$ .

14. The second Fundamental Theorem

The relationship can also be written as:

$\int_a^b f(x),dx=F(b)-F(a)$

where:

$F'(x)=f(x)$

This is often called the Second Fundamental Theorem of Calculus.

It gives us the practical computational rule:

Find an antiderivative, evaluate it at the upper limit, evaluate it at the lower limit, and subtract.

In shorthand:

$\int_a^b f(x),dx=[F(x)]_a^b=F(b)-F(a)$

This is the formula that makes much of elementary integration manageable.

15. Why the limits matter

Consider:

$\int_2^5 f(x),dx$

The lower limit is 2 and the upper limit is 5.

The integral represents accumulated net change from 2 to 5.

If we reverse the limits:

$\int_5^2 f(x),dx$

the sign changes:

$\int_5^2 f(x),dx=-\int_2^5 f(x),dx$

This is another reminder that definite integration is not simply “area.”

It represents signed accumulation.

Areas below the x-axis contribute negatively.

16. The deeper picture

We can now see the entire calculus story more clearly.

Suppose we have a quantity:

$Q(x)$

Its derivative tells us its rate of change:

$Q'(x)=q(x)$

If we know the rate $q(x)$ instead, we can recover the change in $Q$ :

$Q(b)-Q(a)=\int_a^b q(x),dx$

So:

$\boxed{\text{Total change}=\int \text{rate of change}}$

This is perhaps the most useful way to remember the Fundamental Theorem.

17. A universal pattern

The idea appears everywhere.

If we know:

velocity, integrate to obtain displacement.

$\Delta s=\int v(t),dt$

If we know:

acceleration, integrate to obtain change in velocity.

$\Delta v=\int a(t),dt$

If we know:

marginal cost, integrate to obtain change in total cost.

$\Delta C=\int MC(q),dq$

If we know:

marginal revenue, integrate to obtain change in revenue.

$\Delta R=\int MR(q),dq$

If we know:

population growth rate, integrate to obtain accumulated population change.

$\Delta P=\int g(t),dt$

The pattern is always the same:

$\boxed{\text{Accumulated change}=\int\text{rate of change}}$

18. The real meaning of the Fundamental Theorem

The Fundamental Theorem of Calculus is much more than a formula for calculating areas.

It tells us that change and accumulation are two sides of the same mathematical process.

Differentiation asks:

How much is something changing right now?

Integration asks:

How much change has accumulated over an interval?

And the Fundamental Theorem tells us:

$\boxed{\text{Integration and differentiation undo each other}}$

That is the conceptual breakthrough at the heart of calculus.

19. From infinitely many pieces to one simple calculation

Perhaps the most beautiful aspect of the theorem is the computational shortcut it provides.

Integration begins with the idea of adding infinitely many tiny contributions:

$\int_a^b f(x),dx$

Yet, if we can find an antiderivative $F$ , the entire accumulation becomes:

$F(b)-F(a)$

An apparently infinite problem becomes a finite calculation.

That is why the Fundamental Theorem changed mathematics.

It gave us a systematic way to move between:

$\text{infinitesimal change}\quad\text{and}\quad\text{finite accumulation}$

Conclusion

The Fundamental Theorem of Calculus is the bridge connecting the two great ideas of calculus.

Differentiation studies rates of change.

Integration studies accumulation.

The theorem tells us that these are not unrelated mathematical operations. They are deeply connected:

$\frac{d}{dx}\left(\int_a^x f(t),dt\right)=f(x)$

and, if $F'(x)=f(x)$ ,

$\int_a^b f(x),dx=F(b)-F(a)$

Once this connection is understood, integration becomes much more than finding areas.

It becomes a general language for reconstructing totals from rates, changes from marginal quantities, distances from velocities, and accumulated effects from instantaneous processes.

And that is perhaps the central insight of calculus:

If differentiation tells us how things change, integration tells us what all those changes add up to.

Applications of Integration: From Accumulation to Real-World Decisions

August 12, 2026 by Splendid Leave a Comment

Integration is often introduced as the reverse of differentiation. If differentiation tells us how quickly something is changing, integration helps us reconstruct the total quantity accumulated from those changes.

That simple idea makes integration one of the most powerful tools in mathematics.

Integration allows us to calculate areas, distances, volumes, revenues, costs, probabilities, population changes, energy consumption, and many other quantities that are built up continuously from smaller pieces.

1. Integration as Accumulation

Suppose a car is moving at a speed of 60 km/h. After one hour, the distance travelled is easy to calculate:

$Distance=Speed\times Time=60\times1=60\text{ km}$

But what if the speed keeps changing?

Perhaps the car travels at:

$50\text{ km/h},\quad55\text{ km/h},\quad63\text{ km/h},\quad70\text{ km/h},\ldots$

Now there is no single speed that we can simply multiply by time.

Integration solves this problem.

If velocity is a function of time, $v(t)$ , then the distance travelled between $t=a$ and $t=b$ is:

$Distance=\int_a^b v(t),dt$

The integral is essentially adding up an enormous number of tiny distances.

2. Finding Area Under a Curve

One of the most famous applications of integration is finding the area under a curve.

Suppose we have a function:

$y=f(x)$

and want the area between $x=a$ and $x=b$ .

We can divide the region into many thin rectangles.

Each rectangle has approximately:

$Area\approx f(x)\Delta x$

Adding them gives:

$Area\approx\sum f(x)\Delta x$

As the rectangles become infinitely thin, the approximation becomes exact:

$Area=\int_a^b f(x),dx$

This is one of the fundamental meanings of integration.

Why is this useful?

Because many real-world quantities cannot be represented by simple rectangles or constant rates.

For example, integration can calculate:

land area
area of irregular objects
area beneath economic curves
accumulated production
probability distributions
consumer and producer surplus

3. Distance from Velocity

Differentiation and integration are almost mirror images here.

If position is:

$s(t)$

then velocity is:

$v(t)=s'(t)$

Integration reverses this process:

$s(t)=\int v(t),dt$

For example, suppose:

$v(t)=3t^2$

Then:

$s(t)=\int3t^2,dt=t^3+C$

The constant $C$ represents the initial position.

So integration allows us to move from:

rate of movement → total movement

This idea extends far beyond physical motion.

4. Acceleration to Velocity

Acceleration is the rate at which velocity changes.

$a(t)=\frac{dv}{dt}$

Therefore, integrating acceleration gives velocity:

$v(t)=\int a(t),dt$

And integrating velocity gives position:

$s(t)=\int v(t),dt$

So we have a chain:

$Acceleration\xrightarrow{\int}Velocity\xrightarrow{\int}Position$

This is a beautiful example of integration as the reconstruction of accumulated change.

5. Calculating Work and Energy

Integration is extremely important in physics.

Suppose a constant force $F$ moves an object through a distance $d$ .

The work done is:

$W=Fd$

But real forces are often not constant.

For example, the force required to stretch a spring increases as the spring gets longer.

If the force is:

$F(x)$

then the work required to move from $x=a$ to $x=b$ is:

$W=\int_a^bF(x),dx$

Again, integration adds up many tiny contributions:

$dW=F(x),dx$

and therefore:

$W=\int dW$

6. Calculating Volume

Integration can also calculate the volume of objects with curved surfaces.

Imagine slicing an object into extremely thin pieces.

If the cross-sectional area at position $x$ is:

$A(x)$

then a thin slice has approximately:

$dV=A(x),dx$

Adding all the slices gives:

$V=\int_a^bA(x),dx$

This is the basic idea behind many volume calculations.

For example, when a region is rotated around an axis, we can use the disk or washer method.

For rotation around the x-axis:

$V=\pi\int_a^b[f(x)]^2,dx$

Thus integration turns a complicated three-dimensional shape into the accumulation of many simple two-dimensional slices.

7. Economics: Total Cost from Marginal Cost

Integration becomes especially interesting in economics.

Suppose a company’s marginal cost is:

$MC(q)=\frac{dC}{dq}$

This means marginal cost tells us approximately how much additional cost arises from producing one more unit.

If we know the marginal cost function, we can recover total cost by integration:

$C(q)=\int MC(q),dq+C_0$

where $C_0$ represents fixed cost.

For example, suppose:

$MC(q)=20+2q$

Then:

$C(q)=\int(20+2q),dq=20q+q^2+C_0$

Integration has therefore converted:

marginal information → total information

This is one of the most important applications of integration in economics.

8. Total Revenue from Marginal Revenue

The same principle applies to revenue.

Marginal revenue is:

$MR(q)=\frac{dR}{dq}$

Therefore:

$R(q)=\int MR(q),dq+C$

Suppose:

$MR(q)=100-4q$

Then:

$R(q)=100q-2q^2+C$

If revenue is zero when $q=0$ , then $C=0$ .

Therefore:

$R(q)=100q-2q^2$

Integration has reconstructed the company’s total revenue function from its marginal revenue.

9. Consumer Surplus and Producer Surplus

Integration is also used to measure economic welfare.

Suppose demand is represented by:

$P=D(q)$

The area under the demand curve represents the total willingness to pay across units.

If the market price is $P_0$ and quantity purchased is $Q_0$ , consumer surplus can be represented as:

$CS=\int_0^{Q_0}[D(q)-P_0],dq$

Similarly, producer surplus can be represented using the supply curve:

$PS=\int_0^{Q_0}[P_0-S(q)],dq$

This demonstrates an important point:

Integration can turn a curve into an economic quantity with real meaning.

10. Population Growth

Suppose the rate at which a population changes is known.

Let:

$\frac{dP}{dt}=r(t)$

Then the change in population between $t=a$ and $t=b$ is:

$\Delta P=\int_a^b r(t),dt$

Therefore:

$P(b)=P(a)+\int_a^b r(t),dt$

The same principle applies to:

population growth
migration
birth rates
death rates
customer acquisition
employee growth
subscriber growth

Whenever we know a rate of change, integration can help recover the accumulated change.

11. Business: Customer Acquisition

Imagine a company acquires customers at a rate of:

$r(t)=100+20t$

customers per month.

The number of new customers acquired during the first 12 months is:

$N=\int_0^{12}(100+20t),dt$

Therefore:

$N=[100t+10t^2]_0^{12}$ $N=2640$

So the company acquired 2,640 customers during those 12 months, assuming the rate function accurately represents the acquisition process.

This is conceptually the same as calculating distance from velocity.

Customer acquisition rate → total customers acquired

12. Revenue from a Continuous Sales Rate

Suppose a business generates revenue at a continuously changing rate:

$r(t)$

The revenue generated between $t=a$ and $t=b$ is:

$Revenue=\int_a^b r(t),dt$

For example, if a website generates advertising revenue at a rate that changes throughout the day, integration can theoretically calculate the total revenue generated over the entire day.

The same principle applies to:

advertising revenue
subscription revenue
sales
electricity consumption
production
website traffic
transaction volume

13. Probability

Integration plays a fundamental role in probability.

For a continuous random variable with probability density function $f(x)$ , the probability that $X$ lies between $a$ and $b$ is:

$P(a\leq X\leq b)=\int_a^bf(x),dx$

The entire probability distribution must add up to 1:

$\int_{-\infty}^{\infty}f(x),dx=1$

So probability density can be understood as something that is accumulated through integration to produce probability.

This is why integration is central to statistics, data science, finance, and machine learning.

14. Expected Value

Integration can also calculate the expected value of a continuous random variable.

If $f(x)$ is its probability density function, then:

$E[X]=\int_{-\infty}^{\infty}xf(x),dx$

This is essentially a weighted average.

Values that are more probable contribute more heavily to the result.

Expected values are widely used in:

finance
insurance
economics
statistics
risk analysis
machine learning

15. Finance: Continuous Cash Flows

Suppose money is being generated or received continuously at a rate:

$R(t)$

Then the total undiscounted cash flow over a period is:

$C=\int_a^bR(t),dt$

In financial mathematics, we can also incorporate the time value of money.

If the continuous cash-flow rate is $R(t)$ and the continuous discount rate is $r$ , the present value can be represented as:

$PV=\int_a^bR(t)e^{-rt},dt$

Integration therefore allows financial analysts to value streams of continuously occurring cash flows.

16. Electricity and Energy Consumption

Suppose electrical power consumption varies with time.

Power is the rate at which energy is consumed:

$P(t)=\frac{dE}{dt}$

Therefore total energy consumed is:

$E=\int_a^bP(t),dt$

This is particularly useful because electricity demand is rarely constant.

A building might consume different amounts of electricity during:

morning
afternoon
evening
night

Integration adds up these changing consumption rates.

17. Engineering

Engineers use integration constantly.

It appears in:

structural engineering
mechanical engineering
electrical engineering
civil engineering
aerospace engineering
control systems

For example, if a distributed load acts along a beam, integration can determine the total force.

If the load intensity is:

$w(x)$

then total load is:

$F=\int_a^bw(x),dx$

The same idea can be extended to determine moments, stresses, centers of mass, and other engineering quantities.

18. Finding the Center of Mass

Integration can determine the center of mass of an object whose mass is distributed continuously.

For a one-dimensional distribution, the center of mass can be expressed as:

$\bar{x}=\frac{\int_a^bx\rho(x),dx}{\int_a^b\rho(x),dx}$

where $\rho(x)$ represents mass density.

Instead of treating the entire object as one point, we divide it into tiny pieces, determine the contribution of each piece, and integrate.

19. Why Integration Is So Powerful

All these applications may initially appear unrelated.

What does the area under a curve have to do with:

distance travelled?
revenue?
population?
probability?
energy?
economic surplus?

But underneath them is exactly the same mathematical idea.

Integration adds up infinitely many tiny contributions.

Consider these examples:

Small quantity	Integration produces
Velocity × tiny time	Distance
Force × tiny distance	Work
Power × tiny time	Energy
Marginal cost × tiny quantity	Total cost
Marginal revenue × tiny quantity	Total revenue
Growth rate × tiny time	Population change
Probability density × tiny interval	Probability
Revenue rate × tiny time	Total revenue

The formula changes, but the underlying logic remains the same.

20. The Deep Connection with Differentiation

This brings us back to the relationship between differentiation and integration.

Differentiation asks:

How fast is something changing right now?

Integration asks:

How much change has accumulated over an interval?

For example:

$v(t)=\frac{ds}{dt}$

says velocity is the rate of change of position.

Integration reverses the relationship:

$s(b)-s(a)=\int_a^bv(t),dt$

Similarly:

$MC(q)=\frac{dC}{dq}$

while:

$C(Q)-C(0)=\int_0^QMC(q),dq$

And:

$P(t)=\frac{dE}{dt}$

while:

$E(b)-E(a)=\int_a^bP(t),dt$

This is the fundamental pattern:

Rate → Integration → Accumulated quantity

21. Integration Is Continuous Addition

Perhaps the simplest way to understand integration is to think of it as an extension of ordinary addition.

Suppose you earn ₹100 every day for 30 days.

You can simply calculate:

$100+100+\cdots+100=3000$

But suppose your daily earnings are different:

$₹80,₹120,₹95,₹140,\ldots$

You add them individually.

Now imagine that earnings are changing continuously, rather than once per day.

There are infinitely many tiny contributions.

Integration is the mathematical machinery that performs this continuous accumulation:

$Total=\int\text{rate}\times\text{tiny interval}$

That is the heart of integration.

Conclusion

Integration is much more than a technique for finding areas.

It is a general mathematical language for accumulation.

Whenever something is changing continuously and we want to know the total effect of that change, integration becomes a natural tool.

It can transform:

$Velocity\rightarrow Distance$ $Acceleration\rightarrow Velocity$ $Marginal\ Cost\rightarrow Total\ Cost$ $Marginal\ Revenue\rightarrow Total\ Revenue$ $Power\rightarrow Energy$ $Growth\ Rate\rightarrow Total\ Growth$ $Probability\ Density\rightarrow Probability$ $Cash\ Flow\ Rate\rightarrow Total\ Cash\ Flow$

So perhaps the most useful mental model is:

Differentiation breaks change down into a rate. Integration builds countless tiny changes back into a whole.

That is why integration appears everywhere—from physics and engineering to economics, finance, statistics, business, and data science.

And this is also why integration can be viewed as the reverse side of differentiation: differentiation tells us what is happening locally, while integration tells us what those local changes collectively produce.

integration

1. Two seemingly different questions

2. Differentiation starts with accumulation and finds the rate

3. Integration goes in the opposite direction

4. Integration is really accumulation

5. Why does this make intuitive sense?

6. The first Fundamental Theorem

7. But what about the mysterious +C?

8. Definite integration changes the situation

9. Why this theorem is so powerful

10. The theorem connects two worlds

11. A beautiful example: velocity and distance

12. Another example: changing revenue

13. Integration and differentiation are not merely opposite tricks

14. The second Fundamental Theorem

15. Why the limits matter

16. The deeper picture

17. A universal pattern

18. The real meaning of the Fundamental Theorem

19. From infinitely many pieces to one simple calculation

Conclusion

1. Integration as Accumulation

2. Finding Area Under a Curve

Why is this useful?

3. Distance from Velocity

4. Acceleration to Velocity

5. Calculating Work and Energy

6. Calculating Volume

7. Economics: Total Cost from Marginal Cost

8. Total Revenue from Marginal Revenue

9. Consumer Surplus and Producer Surplus

10. Population Growth

11. Business: Customer Acquisition

12. Revenue from a Continuous Sales Rate

13. Probability

14. Expected Value

15. Finance: Continuous Cash Flows

16. Electricity and Energy Consumption

17. Engineering

18. Finding the Center of Mass

19. Why Integration Is So Powerful

20. The Deep Connection with Differentiation

21. Integration Is Continuous Addition

Conclusion

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Calculus 1A: Differentiation

Calculus 1B: Integration

Calculus 1C: Coordinate Systems & Infinite Series