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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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BuyFindarrow_forward

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

  1. (a) Calculate the surface area of the ellipsoid that is generated by rotating an ellipse about its major axis.
  2. (b) What is the surface area if the ellipse is rotated about its minor axis?

(a)

To determine

To find: The surface area of the ellipse by rotating about its major axis.

Explanation

Calculation:

Write the ellipse equation as below.

x2a2+y2b2=1 (1)

The value of x and y are x=acost and y=bsint .

Substitute acost for x and bsint for y in the equation (1).

x2a2+y2b2=1(acost)2a2+(bsint)2b2=1

Differentiate the value of x and y with respect to t .

x=acostdxdt=asinty=bsintdydt=bcost

Calculate the value of (dxdt)2+(dydt)2 .

(dxdt)2+(dydt)2=(asint)2+(bcost)2=a2sin2t+b2cos2t

Substitute 1cos2t for sin2t in the above equation.

(dxdt)2+(dydt)2=a2sin2t+b2cos2t=a2(1cos2t)+b2cos2t=a2a2cos2t+b2cos2t(dxdt)2+(dydt)2=a2+cos2t(b2a2)

The equation to be used is c2=a2b2 .

Substitute c2 for b2a2 in the above equation.

(dxdt)2+(dydt)2=a2+cos2t(b2a2)=a2+cos2t(c2)=a2c2cos2t

Calculate the surface area of the ellipse about major axis.

S=ab2πy(dxdt)2+(dydt)2dt

Substitute bsint for y and a2c2cos2t for (dxdt)2+(dydt)2 in the above equation.

S=ab2πy(dxdt)2+(dydt)2dt

S=0π22π(bsint)a2c2cos2tdt (2)

Take u=ccost .

Differentiate the above equation with respect to t .

u=ccostdu=csintdtdt=ducsint

Find the limits as below.

u=ccos(0)u=c(1)u=cu=ccos(π2)u=c(0)u=0

Therefore, the limit is c0

(b)

To determine

To find: The surface area of the ellipse by rotating about its minor axis.

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