Question

Find the volumes of the solids obtained by rotating the region bounded by the curves y = x and y = x^2 about the following lines.

(a) The x-axis
(b) The y-axis
(c) y = 2

Answer 1

(a) A cross-section is a washer with inner radius
`x^2` and outer radius
`x`.


`V = \int_0^1 \pi\left[(x)^2 - (x^2)^2\right]dx = \int_0^1 \pi(x^2 - x^4) dx = \pi \left[ (1)/(3)x^3 - (1)/(5)x^5\right]_0^1 \\ \\ = \pi \left[(1)/(3) - (1)/(5)\right] = (2)/(15)\pi`

(b)A cross-section is a washer with inner radius
`y` and outer radius
`√(y)`.


`V = \int_0^1 \pi \left[ \left( √(y) \right)^2 - y^2 \right]dy = \int_0^1 \pi(y - y^2)dy = \pi \left[ (1)/(2)y^2 - (1)/(3)y^3\right]_0^1 \\ = \pi \left[ (1)/(2) - (1)/(3) \right] = (\pi)/(6)`

(c) A cross-section is a washer with inner radius
`2-x` and outer radius
`2 - x^2`


`V = \int_0^1 \pi\left[ (2-x^2)^2 - (2-x)^2\right] = \int_0^1 \pi (x^4 - 5x^2 + 4x) dx \\ \\ = \pi \left[ (1)/(5)x^5 - (5)/(3)x^3 + 2x^2 \right]_0^1 = \pi\left[ (1)/(5) - (5)/(3) + 2\right] = (8)/(15)\pi`