Calculus II Study Guide: Applications of Integration and Volumes of Revolution

Applications of Integration

Surface Area of Revolution

When a curve $y = f(x)$ is revolved about the x-axis between $x = a$ and $x = b$, the surface area $S$ of the resulting solid is given by:

• Formula:

$S = 2\pi \int_a^b f(x) \sqrt{1 + (f'(x))^2} \, dx$

• Application: Used to find the surface area of solids of revolution, such as the surface of a vase or a bell.

• Example: Find the surface area generated by revolving $y = x^2$ from $x = 0$ to $x = 1$ about the x-axis.

Trigonometric Identities for Integration

Trigonometric identities are essential tools for simplifying integrals involving trigonometric functions.

• Key Identities:

• $\sin^2(x) + \cos^2(x) = 1$

• $1 + \tan^2(x) = \sec^2(x)$

• $\sin^2(x) = 1 - \cos^2(x)$

• $\cos(2x) = 2\cos^2(x) - 1$

• $\sin(2x) = 2\sin(x)\cos(x)$

• $\cos^2(x) = \frac{1 + \cos(2x)}{2}$

• $\sin^2(x) = \frac{1 - \cos(2x)}{2}$

Application: These identities are used to rewrite integrals in a form that is easier to evaluate, especially in problems involving powers of sine and cosine.

Definite Integrals and Substitution

Evaluating Integrals

Definite integrals are used to calculate areas, volumes, and other quantities. Substitution is a common technique for simplifying integrals.

• Substitution Method: If $u = g(x)$, then $\int f(g(x))g'(x)dx = \int f(u)du$.

• Example: To evaluate $\int x\cos(x^2)dx$, let $u = x^2$, then $du = 2x dx$.

Area Between Curves

Setting Up Area Integrals

The area between two curves $y = f(x)$ and $y = g(x)$ from $x = a$ to $x = b$ is:

• Formula:

$A = \int_a^b [f(x) - g(x)] dx$

• Application: Used to find the area enclosed between curves, such as between $y = x^2$ and $y = x + 2$.

• Example: Find the area between $y = x^2$ and $y = 2x$ from $x = 0$ to $x = 2$.

Area Using Vertical and Horizontal Slices

Sometimes, it is easier to integrate with respect to $y$ rather than $x$, especially when the region is bounded by functions of $y$.

• Formula (with respect to $y$):

$A = \int_{y_1}^{y_2} [x_{right}(y) - x_{left}(y)] dy$

• Example: Find the area between $x = y^2$ and $x = y + 2$ for $y$ in $[0,2]$.

Volumes of Revolution

Disk and Washer Methods

These methods are used to find the volume of a solid generated by revolving a region around an axis.

• Disk Method: Used when the solid has no hole.

• Formula:

$V = \pi \int_a^b [R(x)]^2 dx$

• Washer Method: Used when the solid has a hole (region between two curves).

• Formula:

$V = \pi \int_a^b \left([R_{outer}(x)]^2 - [R_{inner}(x)]^2\right) dx$

• Example: Find the volume generated by revolving the region between $y = x^2$ and $y = x$ about the x-axis.

Shell Method

The shell method is useful when revolving around a vertical or horizontal axis, especially when the region is described in terms of $y$.

• Formula:

$V = 2\pi \int_a^b (radius)(height) dx$

• Example: Find the volume generated by revolving the region between $x = y^2$ and $x = y + 2$ about the y-axis.

Arc Length of Curves

Arc Length Formula

The length $L$ of a curve $y = f(x)$ from $x = a$ to $x = b$ is:

• Formula:

$L = \int_a^b \sqrt{1 + (f'(x))^2} dx$

• Application: Used to find the length of curves, such as the length of $y = x^2$ from $x = 0$ to $x = 1$.

Practice Problems Overview

The provided problems cover:

• Evaluating definite integrals using substitution and trigonometric identities

• Setting up and interpreting area and volume integrals for regions bounded by curves

• Using disk, washer, and shell methods for volumes of revolution

• Calculating arc lengths and surface areas of curves

• Classifying and comparing different integral setups for geometric applications

Key Table: Trigonometric Identities for Integration

Identity	Equation	Application
Pythagorean Identity	$\sin^2(x) + \cos^2(x) = 1$	Simplifies powers of sine and cosine
Tangent-Secant Identity	$1 + \tan^2(x) = \sec^2(x)$	Useful for integrals involving tangent and secant
Double Angle for Cosine	$\cos(2x) = 2\cos^2(x) - 1$	Reduces powers of cosine
Double Angle for Sine	$\sin(2x) = 2\sin(x)\cos(x)$	Product to sum conversion
Power Reduction for Cosine	$\cos^2(x) = \frac{1 + \cos(2x)}{2}$	Integrals of $\cos^2(x)$
Power Reduction for Sine	$\sin^2(x) = \frac{1 - \cos(2x)}{2}$	Integrals of $\sin^2(x)$

Additional info:

• Problems are drawn from Calculus II topics, specifically sections on applications of integration, area, volume, arc length, and surface area.

• Practice problems include both computational and conceptual questions, with diagrams illustrating regions and solids.

• Students should be familiar with setting up integrals for geometric quantities and using substitution and trigonometric identities to evaluate them.