Calculus
posted by Anonymous .
Find the volume of the solid whose base is the region in the xyplane bounded by the given curves and whose crosssections perpendicular to the xaxis 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=8x^2, y=x^2
(a) integral (82x)^2 from 2 to 2 = 192
(b) integral 1/2 pi [(82x^2)/2]^2 from 2 to 2 =256pi/15
(c) integral (82x^2)(sqrt3)(4x^2) from 2 to 2 = 1024(sqrt3)/15
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