Mathematics, 24.07.2019 02:00 deandrehudson18
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
Answers: 1
and outer radius 
.![V = \int_0^1 \pi\left[(x)^2 - (x^2)^2\right]dx = \int_0^1 \pi(x^2 - x^4) dx = \pi \left[ \frac{1}{3}x^3 - \frac{1}{5}x^5\right]_0^1 \\ \\ = \pi \left[\frac{1}{3} - \frac{1}{5}\right] = \frac{2}{15}\pi](/tpl/images/0125/6073/98b32.png)
and outer radius 
.![V = \int_0^1 \pi \left[ \left( \sqrt{y} \right)^2 - y^2 \right]dy = \int_0^1 \pi(y - y^2)dy = \pi \left[ \frac{1}{2}y^2 - \frac{1}{3}y^3\right]_0^1 \\ = \pi \left[ \frac{1}{2} - \frac{1}{3} \right] = \frac{\pi}{6}](/tpl/images/0125/6073/1f149.png)
and outer radius 
![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[ \frac{1}{5}x^5 - \frac{5}{3}x^3 + 2x^2 \right]_0^1 = \pi\left[ \frac{1}{5} - \frac{5}{3} + 2\right] = \frac{8}{15}\pi](/tpl/images/0125/6073/de46b.png)
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Find the volumes of the solids obtained by rotating the region bounded by the curves y = x and y = x...
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