Question

Calculate the volume of the solid obtained by rotating around the x-axis:
a) 0<= x <= 1 and sqrt(x) <= y <= 3
b) x² <= y <= x
c) 0 <= y <= x and x²+y² <= 2
d) y >= x² and x² + y² <= 2
e) 1/x <= y <= 1 and 1<= x <=2
f) x² + (y-2)² <= 1

folks, those exercises I couldn't solve. My answers is different from the book's answers. I hope to see yours. Thank you very much.

Answer 1

Volume of revolution formula:
\[V = \pi \int\limits_{a}^{b} f(x)^2 dx \]
a)
\[\pi \left( \int\limits\limits_{0}^{1} (3)^2dx - \int\limits\limits_{0}^{1} (\sqrt{x})^2 dx\right) =\pi \left( \left[ 9x \right]_0^1 - \left[ \frac{ 1 }{ 2 }x^2 \right]_0^1 \right) =\]\[ \pi \left( \left[ 9 - 0 \right] - \left[ \frac{ 1 }{ 2 }-0 \right] \right) = \frac{17}{2} \pi\]
b)
\[\pi \left(\int\limits_{0}^{1}(x)^2dx-\int\limits_{0}^{1}(x^2)^2dx \right)=\pi \left( \left[ \frac{ 1 }{ 3 }x^3 \right]_0^1 - \left[ \frac{1}{5}x^5\right]_0^1 \right) = \]\[\pi \left( \left[\frac{1}{3}-0\right]-\left[\frac{1}{5}-0\right] \right)= \frac{2\pi}{15}\]
c)\[\pi \left( \int\limits\limits_{0}^{1} (x)^2dx+\int\limits\limits_{1}^{\sqrt{2}}\left(\sqrt{2-x^2}\right)^2dx \right)=\pi\left(\left[\frac{1}{3}x^3\right]_0^1+\left[2x-\frac{1}{3}x^3\right]_1^\sqrt{2}\right)=\]\[\pi \left( \left[\frac{1}{3}-0\right]+\left[\frac{4\sqrt{2}}{3}-\frac{5}{3} \right]\right)=\frac{4\pi}{3}\left(\sqrt{2}-1\right)\]
d) y-axis symetry used
\[2\pi \left(\int\limits\limits_0^1\left(\sqrt{2-x^2}\right)^2dx - \int\limits\limits_0^1\left(x^2\right)^2dx\right) = 2\pi \left( \left[2x-\frac{1}{3}x^3\right]_0^1-\left[\frac{1}{3}x^3\right]_0^1\right)=\]
\[2\pi \left( \left[\frac{5}{3}-0 \right] - \left[\frac{1}{3}-0\right] \right) = \frac{8\pi}{3}\]
e)\[\pi \left( \int\limits\limits\limits\limits_1^2 (1)^2dx - \int\limits_1^2 \left(\frac{1}{x}\right)^2dx\right) = \pi \left( \left[x\right]_1^2-\left[-\frac{1}{x}\right]_1^2 \right)=\] \[ \pi \left( \left[2-1\right]-\left[-\frac{1}{2}+\frac{1}{1}\right]\right)= \frac{\pi}{2}\]
f)Here you should see a torus:
\[V = 2 \pi a \times \pi r^2\]
where
a = diameter of the revolution - in this case the distance between x-axis and the center of the circle,
r = diameter of the circle.
It always works like this - circuit of revolution times surface being revoluted.