AP Calculus: Applications of Integration

This chapter extends the concept of the definite integral beyond finding the area under a single curve to solve a variety of geometric problems. The central idea is to use integration as a tool for summing up an infinite number of small pieces, whether they are thin rectangles, circular disks, or tiny segments of a curve. By the end of this chapter, you will be able to calculate the area between two curves, a direct application of the definite integral.

You will also learn to find the volume of three-dimensional solids. This involves two main approaches: calculating the volume of solids of revolution using the disk and washer methods, and finding the volume of solids with known, uniform cross-sections. Other applications include calculating the precise length of a curve in the plane (arc length) and, for AP Calculus BC, the surface area of a solid of revolution.

This chapter relies heavily on your ability to set up and evaluate definite integrals, making the previous chapter on integration techniques a direct prerequisite. The most challenging part is often not the calculus itself, but the initial setup. Students must correctly visualize the region or solid, choose the appropriate variable of integration (x or y), determine the correct limits of integration, and define the function that represents the area or radius. A mistake in this setup phase is the most common source of error.

Formulas for AP Calculus: Applications of Integration

Area between two curves: $A = \int_{a}^{b} [f(x) - g(x)] \,dx$ Where f(x) is the upper function and g(x) is the lower function on [a, b].
Area between two curves (with respect to y): $A = \int_{c}^{d} [f(y) - g(y)] \,dy$ Where f(y) is the right function and g(y) is the left function on [c, d].
Volume by Disk Method (horizontal axis of revolution): $V = \pi \int_{a}^{b} [R(x)]^2 \,dx$ R(x) is the radius of the disk as a function of x.
Volume by Washer Method (horizontal axis of revolution): $V = \pi \int_{a}^{b} ([R(x)]^2 - [r(x)]^2) \,dx$ R(x) is the outer radius and r(x) is the inner radius.
Volume by Washer Method (vertical axis of revolution): $V = \pi \int_{c}^{d} ([R(y)]^2 - [r(y)]^2) \,dy$ R(y) is the outer radius and r(y) is the inner radius.
Volume with known cross-sections: $V = \int_{a}^{b} A(x) \,dx$ A(x) is the area of the cross-section perpendicular to the x-axis.
Arc Length of a curve: $L = \int_{a}^{b} \sqrt{1 + [f'(x)]^2} \,dx$ For a function y = f(x) from x=a to x=b.
Arc Length of a curve (with respect to y): $L = \int_{c}^{d} \sqrt{1 + [g'(y)]^2} \,dy$ For a function x = g(y) from y=c to y=d.
Arc Length of a parametric curve: $L = \int_{t_1}^{t_2} \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} \,dt$ For a curve defined by x(t) and y(t).