   Chapter 8.2, Problem 12E

Chapter
Section
Textbook Problem

Find the exact area of the surface obtained by rotating the curve about the x-axis.12. y = x 3 6 + 1 2 x ,   1 2 ≤ x ≤ 1

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=x36+12x,12x1 .

The curve is bounded between x=12 and x=1 .

Calculation:

Show the equation of the curve.

y=x36+12x (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(x36+12x)=ddx(x36+12x1)=16(3x2)+12(x2)=x2212x2

Substitute (x2212x2) for dydx , (x36+12x) for y, 12 for a, and 1 for b in Equation (2).

S=1212π(x36+12x)1+(x2212x2)2dx=1212π(x36+12x)1+[(x22)22×12x2×x22+(12x2)2]dx=1212π(x36+12x)1+[x4412+14x4]dx=1212π(x36+12x)x44+14x4+12dx

Simplify the Equation

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