   Chapter 8.2, Problem 9E

Chapter
Section
Textbook Problem

Find the exact area of the surface obtained by rotating the curve about the x-axis.9. y2 = x + 1, 0 ≤ x ≤ 3

To determine

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

Explanation

Given information:

The equation of the curve is y2=x+1,0x3 .

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

Calculation:

Show the equation of the curve.

y2=x+1y=x+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(x+1)=ddx(x+1)12=12(x+1)121=12(x+1)12

dydx=12x+1

Substitute 12x+1 for dydx , x+1 for y, 0 for a, and 3 for b in Equation (2).

S=032πx+11+[12x+1]2dx=032πx+11+14(x+1)dx=032πx+14x+4+14(x+1)dx=032πx+1(4x+4+12x+1)dx

S=03π(4x+5)dx (3)

Consider the function u=4x+5 (4)

Calculate the upper limit of the function u using Equation (4)

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