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Unit 7 Applications of Integrals

Volumes of Revolutions: AP Calculus AB Rundown

This document provides a concise overview of volumes of revolution, a key topic in AP Calculus AB. We will cover the main methods: the Disk Method, the Washer Method, and briefly touch on the Shell Method (though it’s less common on the AB exam).

1. Introduction

Volumes of revolution involve calculating the volume of a 3D solid formed by rotating a 2D region around a line (the axis of revolution). The key is to slice the solid into infinitesimally thin shapes (disks or washers) and integrate their volumes.

2. The Disk Method

The Disk Method is used when the region being revolved is adjacent to the axis of revolution. This means the cross-sections perpendicular to the axis of revolution are solid disks.

Procedure:

1. Sketch the region: Draw the region you’re revolving and the axis of revolution.
2. Identify the radius: Determine the radius, r, of a representative disk. The radius will be a function of either x or y, depending on the axis of revolution. Radius Identification
3. Set up the integral: The volume is found by integrating the area of the disk, $$ A = \pi r^2 $$ , along the axis of revolution.

Formulas:

• Revolution about the x-axis: $$ V = \pi \int_a^b [f(x)]^2 , dx $$ where f(x) is the function defining the curve, and a and b are the limits of integration along the x-axis.

• Revolution about the y-axis: $$ V = \pi \int_c^d [g(y)]^2 , dy $$ where g(y) is the function defining the curve (solved for x in terms of y), and c and d are the limits of integration along the y-axis.

Example:

Find the volume of the solid formed by revolving the region bounded by $$ y = \sqrt{x} $$ , $$ x = 4 $$ , and $$ y = 0 $$ about the x-axis.

$$ V = \pi \int_0^4 (\sqrt{x})^2 , dx = \pi \int_0^4 x , dx = \pi \left[ \frac{1}{2}x^2 \right]_0^4 = \pi \left( \frac{1}{2}(4)^2 - 0 \right) = 8\pi $$

3. The Washer Method

The Washer Method is used when the region being revolved is not adjacent to the axis of revolution. This creates a “hole” in the center of the solid, resulting in washer-shaped cross-sections.

Procedure:

1. Sketch the region: Draw the region you’re revolving and the axis of revolution.
2. Identify the outer and inner radii: Determine the outer radius, R, and the inner radius, r, of a representative washer. Both radii will be functions of either x or y. Radii Identification
3. Set up the integral: The volume is found by integrating the area of the washer, $$ A = \pi (R^2 - r^2) $$ , along the axis of revolution.

Formulas:

• Revolution about the x-axis: $$ V = \pi \int_a^b ([f(x)]^2 - [g(x)]^2) , dx $$ where f(x) is the outer function (further from the axis) and g(x) is the inner function (closer to the axis), and a and b are the limits of integration along the x-axis.

• Revolution about the y-axis: $$ V = \pi \int_c^d ([F(y)]^2 - [G(y)]^2) , dy $$ where F(y) is the outer function (further from the axis) and G(y) is the inner function (closer to the axis), and c and d are the limits of integration along the y-axis.

Example:

Find the volume of the solid formed by revolving the region bounded by $$ y = x^2 $$ and $$ y = x $$ about the x-axis.

First, find the intersection points: $$ x^2 = x \Rightarrow x^2 - x = 0 \Rightarrow x(x-1) = 0 \Rightarrow x = 0, 1 $$ .

$$ V = \pi \int_0^1 (x^2 - (x^2)^2) , dx = \pi \int_0^1 (x^2 - x^4) , dx = \pi \left[ \frac{1}{3}x^3 - \frac{1}{5}x^5 \right]_0^1 = \pi \left( \frac{1}{3} - \frac{1}{5} \right) = \frac{2\pi}{15} $$