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Area Between Two Curves: Moving Beyond the Area Under a Curve

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

If a function is:

y=f(x)

then the area between the curve and the x-axis can be calculated using:

\int_a^b f(x),dx

This idea is powerful, but many real-world problems involve two changing quantities rather than one.

For example:

  • How much profit remains after costs are subtracted from revenue?
  • How much more efficient is one manufacturing process than another?
  • How much consumer benefit exists beyond the market price?
  • How much faster is one vehicle than another over time?

These questions compare two functions.

This leads to one of the most useful applications of integration:

Finding the area between two curves.

Starting with a simple picture

Suppose we have two functions:

y=f(x)

and

y=g(x)

The shaded region lies between them.

Instead of measuring the distance from the x-axis to a curve, we measure the distance from one curve to another.

The fundamental idea

At any value of x, the vertical distance between the curves is:

f(x)-g(x)

If we add all these tiny vertical distances between a and b, we obtain the total enclosed area.

Therefore:

\boxed{\text{Area}=\int_a^b[f(x)-g(x)],dx}

The upper function always comes first.

The lower function is always subtracted.

Example 1: A parabola and a straight line

Suppose:

f(x)=x+2

and

g(x)=x^2

between:

x=0

and

x=2

The area is:

\int_0^2[(x+2)-x^2],dx

Simplifying:

=\int_0^2(x+2-x^2),dx

Integrating:

=\left(\frac{x^2}{2}+2x-\frac{x^3}{3}\right)_0^2

Evaluating:

=\left(2+4-\frac{8}{3}\right)-0

Therefore:

=\frac{10}{3}

Why subtract the curves?

Think of two runners.

The upper curve represents Runner A.

The lower curve represents Runner B.

The difference:

f(x)-g(x)

tells us how much farther Runner A is ahead at each instant.

Integration adds those differences over an interval.

A business example

Suppose:

Revenue:

R(q)=100q

Cost:

C(q)=20q+0.5q^2

Profit is the difference:

P(q)=R(q)-C(q)

Therefore:

P(q)=80q-0.5q^2

Integration allows us to calculate accumulated profit over a production interval.

The mathematics is identical to finding the area between two curves.

Consumer surplus

Economists frequently use the area between curves.

Suppose:

Demand:

P_d=100-Q

Market price:

P=40

Consumer surplus is:

\int_0^{60}[(100-Q)-40],dQ

The shaded area represents the additional value consumers receive beyond the market price.

A physics example

Suppose two cars travel according to different velocity functions.

The difference:

v_1(t)-v_2(t)

measures the speed advantage at every instant.

Integrating that difference tells us the total advantage accumulated over time.

The key insight

Finding the area under one curve measures accumulation.

Finding the area between two curves measures comparative accumulation.

Instead of asking:

How much?

we ask:

How much more?

Conclusion

The area between two curves extends one of the most important ideas in calculus.

Integration no longer measures a quantity relative to zero.

Instead, it measures the accumulated difference between two changing quantities.

This simple idea lies behind profit analysis, consumer surplus, engineering optimization, and many other real-world applications.

DavidsonNext: AP® Calculus: Challenging Concepts from Calculus AB & Calculus BC

DavidsonNext: AP® Calculus: Challenging Concepts from Calculus AB & Calculus BC

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