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

Question

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

Answer 1

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.