   Chapter 8.2, Problem 24E

Chapter
Section
Textbook Problem

Use either a CAS or a table of integrals to find the exact area of the surface obtained by rotating the given curve about the x-axis.24. y = x 2 + 1 , 0 ≤ x ≤ 3

To determine

To find: The exact area of the surface obtained by rotating the curve about x-axis using table of integrals.

Explanation

Given information:

The Equation of the curve y=x2+1,0x3 .

The curve is bounded between x=0 and x=3 .

Calculation:

Show the equation of the curve.

y=x2+1 (1)

Calculate the area of the surface obtained by rotating the curve about x-axis using the relation:

S=ab2πy1+(dydx)2dx (2)

Here, S is the area of the surface obtained by rotating the curve about x-axis and

axb .

Differentiate both sides of Equation (1) with respect to x.

dydx=ddx(x2+1)=ddx[(x2+1)12]=12(x2+1)121(2x)=xx2+1

Substitute xx2+1 for dydx , x2+1 for y, 0 for a, and 3 for b in Equation (2).

S=032π(x2+1)1+(xx2+1)2dx=2π03x2+11+x2x2+1dx=2π03x2+1x2+1+x2x2+1dx=2π03x2+12x2+1x2+1dx

S=2π032x2+1dx=2π032(x2+12)dx=22π03x2+(12)2dx (3)

Show the formula for integrals

a2+u2du=u2a2+u2+a22ln(

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