Find the volume between y=1-x^2 and y=0 on the interval -1 to 1.?

Question

Rotated along the x-axis.

Answer 1

volume
= 2pi ∫(1-x^2)^2dx, x from 0 to 1
= 2pi ∫1-2x^2+x^4 dx, x from 0 to 1
= 2pi(1 -2/3 + 1/5)
= 16pi/15
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Remember: y^2 = (1-x^2)^2 = 1-2x^2+x^4, and use symmetry to simplify your calculation.

Answer 2

The formula is : integer between - 1 and 1 of :
pi * y² dx
y² = (1 - x²)² = 1 - 2x² + x^4

the primitive function is : x - 2/3 x^3 + 1/5 x^5

then the volume is :

pi [(1 - 2/3 + 1/5) - (-1 + 2/3 - 1/5)] = pi * [2 - 4/3 + 2/5] = (16 pi)/15

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