   Chapter 8.2, Problem 8E

Chapter
Section
Textbook Problem

Find the exact area of the surface obtained by rotating the curve about the x-axis.8. y = 5 − x , 3 ≤ x ≤ 5

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 y=5x,3x5 .

The curve is bounded between x=3 and x=5 .

Calculation:

Show the equation of the curve.

y=5x (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(5x)=ddx(5x)12=12(5x)121=12(5x)12

dydx=125x

Substitute 125x for dydx , 5x for y, 3 for a, and 5 for b in Equation (2).

S=352π5x1+[125x]2dx=352π5x1+14(5x)dx=352π5x204x+14(5x)dx=352π5x(214x25x)dx

S=03π(214x)dx (3)

Consider the function u=214x (4)

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

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