Chapter 8.2, Problem 12E

### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336

Chapter
Section

### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336
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,12â‰¤xâ‰¤1 .

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=âˆ«ab2Ï€y1+(dydx)2dx (2)

Here, S is the area of the surface obtained by rotating the curve about x-axis and aâ‰¤xâ‰¤b .

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

dydx=ddx(x36+12x)=ddx(x36+12xâˆ’1)=16(3x2)+12(âˆ’xâˆ’2)=x22âˆ’12x2

Substitute (x22âˆ’12x2) for dydx , (x36+12x) for y, 12 for a, and 1 for b in Equation (2).

S=âˆ«1212Ï€(x36+12x)1+(x22âˆ’12x2)2dx=âˆ«1212Ï€(x36+12x)1+[(x22)2âˆ’2Ã—12x2Ã—x22+(12x2)2]dx=âˆ«1212Ï€(x36+12x)1+[x44âˆ’12+14x4]dx=âˆ«1212Ï€(x36+12x)x44+14x4+12dx

Simplify the Equation

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