   Chapter 10.3, Problem 75E

Chapter
Section
Textbook Problem

# Surface Area Give the integral formulas for the areas of the surfaces of revolution formed when a smooth curve C is revolved about (a) the x-axis and (b) the y-axis.

(a)

To determine

The integral formula for the area of surface of revolution formed when a smooth curve C is revolved about the x-axis.

Explanation

If a smooth curve C given by x=f(t) and y=g(t) does not cross itself on an interval atb, then the area S of the surface of revolution about the x-axis is given by:

S=2πabg(t)(dxdt)2+(dydt)2dt

For example:

Consider the parametric equations:

x=θ+sinθ and y=θ+cosθ

Whose area of revolution is required form the interval 0θπ2 revolved around x-axis.

Now, differentiate x with respect to θ, to get:

dxdθ=1+cosθ …… (1)

And, differentiate y with respect to θ, to get:

dydθ=1sinθ …… (2)

Now, If a smooth curve C given by x=f(t) and y=g(t) does not cross itself on an interval atb, then the area S of the surface of revolution about the x-axis is given by:

S=2πabg(t)(dxdt)2+(dydt)2dt

Since, the provided interval is 0θπ2, so substitute 1+cosθ for dxdθ from equation (1), 1sinθ

(b)

To determine

The integral formula for the area of surface of revolution formed when a smooth curve C is revolved about the y-axis.

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

#### let f(x) = x 1, g(x) = x+1, and h(x) = 2x3 1. Find the rule for each function. 10. g f

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

#### Expand each expression in Exercises 122. (2y+3)(y+5)

Finite Mathematics and Applied Calculus (MindTap Course List)

#### Let and g(x) = x + 3. Then (g ∘ h)(x) = ______.

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 