5.17 Areas with the y-axis and volumes of revolution Quiz

1. Which gives the area between $y=x^{2}$ ($x\ge0$) and the $y$-axis for $0\le y\le 4$?

• A. $\int_0^2 x^{2}\,dx$
• B. $\int_0^4\sqrt{y}\,dy$
• C. $\int_0^4 y^{2}\,dy$
• D. $\pi\int_0^4 y\,dy$

2. The region under $y=x$ for $0\le x\le 3$ is rotated through $2\pi$ about the $x$-axis. Find the volume.

• A. $9\pi$
• B. $\frac{9\pi}{2}$
• C. $27\pi$
• D. $3\pi$

3. The curve $y=e^{x}$ between $y=1$ and $y=e$ is rotated through $2\pi$ about the $y$-axis. Which integrand, with respect to $y$, gives the volume?

• A. $\pi e^{2x}$
• B. $\pi\ln(y^{2})$
• C. $\pi(\ln y)^{2}$
• D. $\pi y^{2}$

4. The region between $y=2$ and $y=x^{2}+1$ for $-1\le x\le 1$ is rotated through $2\pi$ about the $x$-axis. Which integral gives the volume?

• A. $\pi\int_{-1}^{1}\left(2-(x^{2}+1)\right)^{2}dx$
• B. $\pi\int_{-1}^{1}\left((x^{2}+1)^{2}-4\right)dx$
• C. $\pi\int_{-1}^{1}\left(4-(x^{4}+1)\right)dx$
• D. $\pi\int_{-1}^{1}\left(4-(x^{2}+1)^{2}\right)dx$

5. A vase is modelled by rotating $x=\sqrt{y}$ about the $y$-axis. What is the volume of water when the depth is $h$?

• A. $\pi h$
• B. $\frac{\pi h^{2}}{2}$
• C. $\frac{2\pi h^{3/2}}{3}$
• D. $\pi h^{2}$