Answer to Question #167688 in Calculus for Poojashree K

Question #167688

Find the volume of the solid inside the surface r

2 + z

2 = 4 and outside the surface r = 2 cos θ.


1
Expert's answer
2021-03-01T17:42:51-0500

Solution.

In cylindrical coordinates, we can express the cylinder as

r=2cosθ,0θπr=2\cos{\theta}, 0\leq\theta\leq\pi


and the sphere as


r2+z2=4.r^2+z^2=4.

Then, z=±4r2.z=\pm \sqrt{4-r^2}.

So,

V=0π02cosθ4r24r2rdzdrdθ==0π02cosθ204r2rdzdrdθ==0π02cosθ2r4r2drdθ=V=\int\limits_0^π\int\limits_0^{2\cos{\theta}}\int\limits_{-\sqrt{4-r^2}}^{\sqrt{4-r^2}}rdzdrd\theta= \newline =\int\limits_0^π\int\limits_0^{2\cos{\theta}}2\int\limits_{0}^{\sqrt{4-r^2}}rdzdrd\theta= \newline =\int\limits_0^π\int\limits_0^{2\cos{\theta}}2r\sqrt{4-r^2}drd\theta= \newline=0π(23(4r2)32)02cosθdθ==\int\limits_0^π(-\frac{2}{3}(4-r^2)^{\frac{3}{2}})|_0^{2\cos{\theta}}d\theta= \newline=230π(88sin3θ)dθ==\frac{2}{3}\int\limits_0^π(8-8\sin^3\theta )d\theta= \newline=1630π(1sin3θ)dθ==\frac{16}{3}\int\limits_0^π(1-\sin^3\theta )d\theta==163140π(43sinθ+sin3θ)dθ==\frac{16}{3}\cdot \frac{1}{4}\int\limits_0^π(4-3\sin\theta+\sin{3\theta} )d\theta==43(4θ+3cosθ13cos3θ)0π==\frac{4}{3}(4\theta+3\cos\theta-\frac{1}{3}\cos{3\theta} )|_0^π==43(4π3+133+13)==43(4π163)==16π3649.=\frac{4}{3}(4π-3+\frac{1}{3}-3+\frac{1}{3})= \newline =\frac{4}{3}(4π-\frac{16}{3})= \newline =\frac{16π}{3}-\frac{64}{9}.

Answer. 16π3649.\frac{16π}{3}-\frac{64}{9}.


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