Using polar coordinates, find the volume of the solid
Using polar coordinates, find the volume of the solid bounded by the paraboloid: $$ z=x^2+y^2, $$ the cylinders: $$ x^2+y^2=x,\qquad x^2+y^2=2x, $$ and the plane $z=0$.
Explanation of Method
Both cylinders are circles through the origin centered on the positive $x$-axis, with polar equations $r=\cos\theta$ and $r=2\cos\theta$. The base region is the annular-shaped strip between them, and the paraboloid's height $z=r^2$ makes the volume integral straightforward in polar form.
- Convert both cylinders to polar form.
- Set up the volume integral over the strip $\cos\theta\leqslant r\leqslant2\cos\theta$.
- Evaluate using the reduction formula for $\int\cos^4\theta\,d\theta$.
Step 1 · Convert the cylinders to polar form
$x^2+y^2=x\Rightarrow r=\cos\theta$; $x^2+y^2=2x\Rightarrow r=2\cos\theta$. Both require $\cos\theta\geqslant0$, i.e. $\theta\in\left[-\frac\pi2,\frac\pi2\right]$.
Step 2 · Set up the volume integral
$$ V = \int_{-\pi/2}^{\pi/2}\!\int_{\cos\theta}^{2\cos\theta}r^2\cdot r\,dr\,d\theta = \int_{-\pi/2}^{\pi/2}\!\int_{\cos\theta}^{2\cos\theta}r^3\,dr\,d\theta. $$
Step 3 · Evaluate the inner integral
$$ \int_{\cos\theta}^{2\cos\theta}r^3\,dr = \frac{(2\cos\theta)^4-\cos^4\theta}4 = \frac{15\cos^4\theta}4. $$
Step 4 · Evaluate the outer integral
Using $\cos^4\theta=\dfrac{3+4\cos2\theta+\cos4\theta}8$:
$$ \int_{-\pi/2}^{\pi/2}\cos^4\theta\,d\theta = \frac18\left[3\theta+2\sin2\theta+\frac{\sin4\theta}4\right]_{-\pi/2}^{\pi/2} = \frac18\cdot3\pi = \frac{3\pi}8. $$
$$ V = \frac{15}4\cdot\frac{3\pi}8 = \frac{45\pi}{32}. $$
Additional Notes
- This derivation matches the source exactly.
- Verified symbolically: direct evaluation of the nested integral gives exactly $\dfrac{45\pi}{32}$.
Result
The volume is: