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Q 1/10
Score 0
Which statement best describes the region for $\displaystyle \int_{0}^{\pi/3}\int_{0}^{2\cos\theta} r\,dr\,d\theta$ and its area?
300
Sector $0\le \theta\le \pi/3$, $r\le 2$; area $=\tfrac{2\pi}{3}$
Disk of radius 2; area $=4\pi$
Sector-like region $r\le 2\cos\theta$ with $0\le \theta\le \tfrac{\pi}3$; area $=\displaystyle \frac{\sqrt{3}}{4} + \frac{\pi}{3}$.
Triangle with vertices $(0,0),(1,\sqrt{3}),(-1,\sqrt{3})$; area $=\sqrt{3}$
Q 2/10
Score 0
For $u=x^2-y,\ v=y^2-x$, compute $\left|\dfrac{\partial(x,y)}{\partial(u,v)}\right|$ at $(x,y)=(1,1)$.
300
$1/4$
$1$
$1/2$
$1/3$
10 questions
Q.
Which statement best describes the region for $\displaystyle \int_{0}^{\pi/3}\int_{0}^{2\cos\theta} r\,dr\,d\theta$ and its area?
1
300 sec
Q.
For $u=x^2-y,\ v=y^2-x$, compute $\left|\dfrac{\partial(x,y)}{\partial(u,v)}\right|$ at $(x,y)=(1,1)$.
2
300 sec
Q.
Find the tangent plane to $z=\ln(x^2+2y)$ at $(1,1,\ln 3)$. Which is correct?
3
300 sec
Q.
Let $C$ be the ellipse $\frac{x^2}{4}+y^2=1$ oriented counterclockwise. For $\mathbf{F}=\langle xy,\ x-y^2\rangle$, compute $\oint_C \mathbf{F}\cdot d\mathbf{r}$.
4
300 sec
Q.
For $\mathbf{F}=\langle ax,\ y,\ -(a+1)z\rangle$, which statement is TRUE?
5
300 sec
Q.
For the solid cone $x^2+y^2\le z^2$, $0\le z\le 2$ with density $\rho=z$, compute $I_z=\iiint r^2\rho\,dV$.
6
300 sec
Q.
Solid hemisphere $x^2+y^2+z^2\le 1$, $z\ge 0$, density $\rho=1+z$. The $\bar z$ of center of mass is:
7
300 sec
Q.
For $\mathbf{F}=\langle 2xy+\sin y,\ x^2+y+\cos x\rangle$ on $\mathbb{R}^2$, which is TRUE?
8
300 sec
Q.
Let $C$ be $\dfrac{x^2}{9}+\dfrac{y^2}{4}=1$ oriented CCW. Evaluate $\displaystyle \oint_C (x\,dy - y\,dx)$.
9
300 sec
Q.
Volume of the spherical cap of $x^2+y^2+z^2\le 4$ cut by $z\ge 1$ is:
