Calculus

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Find the volume of the solid whose base is the region in the xy-plane bounded by the given curves and whose cross-sections perpendicular to the x-axis are (a) squares, (b) semicircles, and (c) equilateral triangles.

for y=x^2, x=0, and y=0
(a) integral (x^2)^2 from 0 to 2=32/5
(b) integral 1/2 pi (1/2x^2)^2 from 0 to 2 = 4pi/5
(c) integral 1/2x^2(1/2)(x^2)(sqrt(3)) from 0 to 2 = 8(sqrt(3))/5

for y=sqrt(x), x=0, x=16, and y=0
(a) integral (sqrt(x))^2 from 0 to 16= 128
(b) integral 1/2pi[(1/2)(sqrtx)]^2 from 0 to 16 = 16 pi
(c) integral 1/2 (sqrtx)(1/2)(sqrt(x))(sqrt(3))=32sqrt(3)

for y=8-x^2, y=x^2
(a) integral (8-2x)^2 from -2 to 2 = 192
(b) integral 1/2 pi [(8-2x^2)/2]^2 from -2 to 2 =256pi/15
(c) integral (8-2x^2)(sqrt3)(4-x^2) from -2 to 2 = 1024(sqrt3)/15

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