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Volume of a rorating body, rotating y=a*x^2 along y-axis

ÀÛ¼ºÀÚ Uploader : chocopi ÀÛ¼ºÀÏ Upload Date: 2019-08-12º¯°æÀÏ Update Date: 2019-08-13Á¶È¸¼ö View : 246

The volume of the rotating body obtained by rotating y=a*x^2 along y-axis is calculated as follows.
(The radius of the rotating body is equal to x.)

Since x=(y/a)^(1/2),

V = ¡ò¥ðx^2 dy = ¡ò¥ð(y/a) dy = (¥ð/a)¡òy dy = ¥ð/(2a)*y^2

If the integral range is from y1 to y2,

V = ¥ð/(2a)*(y2^2 - y1^2)

Expressing the integral range in terms of x,

y2 = (a*x2)^2, y1 = (a*x1)^2

Then,

V = a*¥ð/2*(x2^4 - x1^4)


*** Âü°í¹®Çå[References] ***

V = a*¥ð/2*(x2^4 - x1^4)
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