超難難題MATH

The curve intersects the x - axis at (0,0) and (a, 0), so integrating from 0 to a.
When revolving about the x - axis, we use the disc method.
So volume = ∫ πy^2 dx = ∫ π ( ax - x^2)^2 dx = ∫ π (a^2x^2 + x^4 - 2ax^3)dx
= π[ a^2x^3/3 + x^5/5 - ax^4/2] from x = 0 to x = a
= π [ a^5/3 + a^5/5 - a^5/2] = πa^5(1/3 + 1/5 - 1/2) = πa^5/30.
When revolving about the y - axis, we use the shell method.
So volume = ∫ 2πxy dx = ∫ 2πx(ax - x^2) dx = ∫ 2π(ax^2 - x^3) dx
= 2π [ ax^3/3 - x^4/4] from x = 0 to x = a
= 2π ( a^4/3 - a^4/4) = πa^4/6
Since the 2 volumes are the same,
πa^4/6 = πa^5/30
a = 30/6 = 5.

volume = ∫ πy^2 dx = ∫ π ( ax - x^2)^2 dx = ∫ π (a^2x^2 + x^4 - 2ax^3)dx
= π[ a^2x^3/3 + x^5/5 - ax^4/2] from x = 0 to x = a
= π [ a^5/3 + a^5/5 - a^5/2] = πa^5(1/3 + 1/5 - 1/2) = πa^5/30.
revolving about the y-axis,so I chose the shell method.
volume = ∫ 2πxy dx = ∫ 2πx(ax - x^2) dx = ∫ 2π(ax^2 - x^3) dx
= 2π [ ax^3/3 - x^4/4] from x = 0 to x = a
= 2π ( a^4/3 - a^4/4) = πa^4/6
Since the two volumes are the same,
πa^4/6 = πa^5/30
a = 30 divide by 6 =5

I hope I can help you with this answer
I didn't copy anybody's
I really didn't copy not fake not lie

For revolving about x-axis

Vol = ∫[0~a] π (ax - x²)² dx

2014-03-05 18:13:01 補充：
For revolving about y-axis

Vol
= ∫[0~a²/4] π [ (a/2 + √(a²/4 - y) )² - (a/2 - √(a²/4 - y) )² ] dy

but the form of A² - B² can be factorized and be a nice form.

{I wish I can do for you, but I am lack of time currently...}

2014-03-05 22:50:36 補充：
無錯，有些時候 shell method 比 disc method 簡單。

wy 大師答得很好～