Solid of revolution question

G

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What is the volume generated by rotating the ellipse with equation

x^2/4 + y^2/9 = 1 about a) the x-axis, b) the y axis?

My answer is:

a) pi * the integral of f(x)^2 dx right? bounded by -2 and 2?

b) don't know how to do this one...

Pls help...
 
x^2/a^2 + y^2/b^2 = 1

x^2/4 + y^2/9 = 1

ellipse has x-axis from -2 to 2, y-axis from -3 to 3

solving for x^2 and y^2 ...

y^2 = 9[1 - (x^2)/4]
x^2 = 4[1 - (y^2)/9]

rotate about the x-axis ...

V = (pi)*INT{-2 to 2} y^2 dx

using symmetry and substituting ...

V = 2(pi)*INT{0 to 2} 9[1 - (x^2)/4] dx

V = 18(pi)*INT{0 to 2} [1 - (x^2)/4] dx

rotate about the y-axis ...

V = (pi)*INT{-3 to 3} x^2 dy

using symmetry and substituting ...

V = 2(pi)*INT{0 to 3} 4[1 - (y^2)/9] dy

V = 8(pi)*INT{0 to 3} [1 - (y^2)/9] dy
 
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