Use a spherical coordinate integral to find the volume of the given solid.
-the solid enclosed by the cardioid of revolution $\varrho = 10 + 7 \cos \varphi , \mathrm { z } \geq 0$
A) $\frac { 10503 } { 2 } \pi$
B) $\frac { 10503 } { 4 } \pi$
C) $\frac { 3501 } { 2 } \pi$
D) $3501 \pi$

Question

Quizplus · Accepted Answer

We can start by finding the limits of integration for $\varphi$, which will be $0\leq\varphi\leq2\pi$. For $	heta$, since the cardioid is symmetric around the $z$-axis, we can choose any range that covers the entire solid. Let's choose $0\leq	heta\leq \pi$.

Now, we need to express $\varrho$ in terms of $r$, $	heta$, and $\varphi$. Using the relationships $x = r\sin\varphi\cos	heta$, $y = r\sin\varphi\sin	heta$, and $z = r\cos\varphi$, we have:

\begin{align*}
r\sin\varphi\cos	heta &= \varrho\sin\varphi\cos	heta \
&= (10 + 7\cos\varphi)\sin\varphi\cos	heta \
&= 10r\sin\varphi\cos	heta + 7r\sin\varphi\cos	heta\cos^2\varphi
\end{align*}

Similarly, we can express $r\sin\varphi\sin	heta$ and $r\cos\varphi$ in terms of $\varrho$, using the relationships:

$$r\sin\varphi\sin	heta = \varrho\sin\varphi\sin	heta = (10 + 7\cos\varphi)\sin\varphi\sin	heta$$

$$r\cos\varphi = \varrho\cos\varphi = (10 + 7\cos\varphi)\cos\varphi$$

Simplifying each of these expressions, we get:

$$r = 10 + 7\cos\varphi$$

$$\sin\varphi\cos	heta = \frac{r}{10}(1-\cos^2\varphi)$$

$$\sin\varphi\sin	heta = \frac{r}{10}\sin\varphi$$

$$\cos\varphi = \frac{r}{10}$$

Now, we can set up the integral:

$$\iiint dV = \int_0^{2\pi}\int_0^\pi\int_0^{10+7\cos\varphi}r^2\sin\varphi\,dr\,d\varphi\,d	heta$$

Evaluating this integral, we get:

\begin{align*}
\int_0^{2\pi}\int_0^\pi\int_0^{10+7\cos\varphi}r^2\sin\varphi\,dr\,d\varphi\,d	heta &= \int_0^{2\pi}\int_0^\pi\left[\frac{1}{3}r^3\sin\varphiight]_0^{10+7\cos\varphi}\,d\varphi\,d	heta \
&=\int_0^{2\pi}\int_0^\pi\frac{1}{3}(10+7\cos\varphi)^3\sin\varphi\,d\varphi\,d	heta \
&= \frac{3501}{2}\pi
\end{align*}

Therefore, the answer is C, $\frac{3501}{2}\pi$.

Use a Spherical Coordinate Integral to Find the Volume of the Given