Area Under the Curve

Understanding area under a curve is a foundational concept in AP Calculus BC. It appears frequently on the AP exam, especially in Free Response Questions (FRQs) involving applications of integrals, accumulation functions, and modeling.

This topic builds on the Fundamental Theorem of Calculus and leads directly to later BC topics such as volumes, arc length, and differential equations.

What “Area Under the Curve” Means

Given a continuous function f(x) on a closed interval [a,b]:

$$\int_{a}^{b}f(x)dx$$

represents the net area between the graph of f(x) and the x-axis.

• Areas above the x-axis are positive

• Areas below the x-axis are negative

If you want total geometric area, you must take absolute values or split intervals.

Fundamental Theorem of Calculus (FTC)

The FTC provides a powerful tool for computing area:

$$\int_{a}^{b}f(x)dx=F(b)-F(a)$$

where $$F^\prime(x) = f(x)$$

This is the basis for almost all area computations on the AP test.

Area Between a Curve and the x-axis

If $f(x)\ge 0$ on [a,b], then the area is:

$$A=\int_{a}^{b}f(x)dx$$

Example

$$\int_{0}^{4}(8-x)dx$$

$$=8x-\frac{x^{2}}{2}\big|_{0}^{4}=(32-\frac{16}{2})-0=24$$

Area and Differential Equations

Sometimes area is used to find:

• particle movement

• solution boundary conditions

• change in population

• work/force modeling

Example

Velocity:

$$v(t)=t^2-4t+3$$

Distance traveled:

$$\int v(t)dt$$

If below x-axis, distance requires absolute value.