Use a double integral to find the volume of the solid shown in the figure.
A)5,996
B)5,994
C)6,000
D)6,008

Question

Use a double integral to find the volume of the solid shown in the figure. 
A)5,996 
B)5,994 
C)6,000 
D)6,008

Quizplus · Accepted Answer

We can use the double integral to find the volume of the solid:

$$V = \iint_D f(x,y) \,dA$$

In this case, $f(x,y) = 4 - x^2 - y^2$, which is the height of the solid at each point $(x,y)$ in the region $D$:

$$D = \{(x,y) \mid x^2 + y^2 \leq 4, x \geq 0, y \geq 0\}$$

We can convert this to polar coordinates to simplify the integral:

$$\begin{aligned} V &= \iint_D (4 - x^2 - y^2) \,dA \ &= \int_0^{\pi/2} \int_0^2 (4 - r^2) r \,dr\,d	heta \ &= \int_0^{\pi/2} (2r^2 - \frac{1}{3}r^4) \Big|_{r=0}^{r=2} \,d	heta \ &= \int_0^{\pi/2} (8 - \frac{32}{3}) \,d	heta \ &= \frac{16}{3} \pi \end{aligned}$$

So the volume of the solid is approximately 6,000 cubic units, which means the answer is C.

Use a Double Integral to Find the Volume of the Solid