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.
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Areas above the x-axis are positive
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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 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:
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particle movement
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solution boundary conditions
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change in population
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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.