Posted by **Taeyeon** on Friday, March 9, 2012 at 12:45pm.

2. Let R be the region of the first quadrant bounded by the x-axis and the cuve y=2X-X^2

a. Find the volume produced when R is revolved around the x-axis

b. Find the volume produced when R is revolved around the y-axis

- calculus -
**Reiny**, Friday, March 9, 2012 at 1:00pm
the x-intercepts are 0 and 2

a) Vol = π ∫ (2x-x^2)^2 dx from 0 to 2

= π∫(4x^2 - 4x^3 + x^4) dx from 0 to 2

the rest is straighforward

b) this is is a little harder.

Solve the function for x

x^2 - 2x + y = 0

x = (2 ± √(4 - 4y) )/2

so the radius = 2 + √(4-4y) - (2 - √(4-4y))

= 2√(4-4y)

so radius^2 = 4(4-4y) = 16 - 16y

volume = π∫(16-16y) dy from 0 to 1 , (the max height of the parabola is 1)

again , the rest is easy.

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