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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336
Textbook Problem

  1. (a) The ellipse x 2 a 2 + y 2 b 2 = 1 a > b is rotated about the x-axis to form a surface called an ellipsoid, or prolate spheroid. Find the surface area of this ellipsoid.
  2. (b) If the ellipse in part (a) is rotated about its minor axis (the y-axis), the resulting ellipsoid is called an oblate spheroid. Find the surface area of this ellipsoid.

a)

To determine

To find: the surface area of an ellipsoid about x-axis.

Explanation

Given:

The equation of the ellipse is x2a2+y2b2=1,a>b

Calculation:

The equation of the ellipse is,

x2a2+y2b2=1 (1)

Differentiate both sides of the Equation (1)

xa2+y(dydx)b2=0y(dydx)b2=xa2dydx=b2xa2y

Calculate the ds using formula:

ds=1+(dydx)2dx (2)

Substitute (b2xa2y) for dydx in Equation (2).

ds=1+(b2xa2y)2dx=1+b4x2a4y2dx=a4y2+b4x2a4y2dx=a4b2(1x2a2)+b4x2a4b2(1x2a2)dx

Simplify:

=a4b2a2b2x2+b4x2a4b2a2b2x2dx=b2(a4a2x2+b2x2)b2(a4a2x2)dx=a4x2(a2b2)a2(a2x2)dx=a4x2(a2b2)aa2x2dx

Rearrange Equation (1).

y2b2=1x2a2=a2x2a2y2=b2a2(a2x2)y=baa2x2

The ellipsoid’s surface area is twice the area generated by rotating the first-quadrant portion of the ellipse about the x-axis.

Calculate the surface area of an ellipsoid’s using the formula:

S=2ab2πyds (3)

Substitute 0 for a, a for b, baa2x2 for y, and a4x2(a2b2)aa2x2dx for ds in Equation (3).

S=20a2πbaa2x2a4x2(a2b2)aa2x2dx=4πba20aa4x2(a2b2)dx=4πba20a(a4x2(a2b2))dx=4πba20a(a4xa2b2)dx (4)

Consider u=xa2b2 (5)

Differentiate both sides of the Equation (5)

b)

To determine

To find: the surface area of an ellipsoid about y-axis.

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