Double Integrals in Polar Coordinates

Use polar coordinates to find the volume of the solid between the cone $z=f(x,y)=\sqrt{x^2+y^2}$ and the paraboloid $z=g(x,y)=\frac{1}{8}\left(x^2+y^2\right) + 2$.

Consider the iterated integral $I = \int_{-3}^{0} \int_{-\sqrt{9-y^2}}^{0} \frac{y}{x^2 + y^2+1} \, dx \, dy.$

1. Sketch (and label) the region of integration.

2. Convert the given iterated integral to one in polar coordinates.

3. Evaluate the iterated integral in (b).

4. State one possible interpretation of the value you found in (c).

Let $D$ be the region that lies inside the unit circle in the plane.

1. Set up and evaluate an iterated integral in polar coordinates whose value is the area of $D$.

2. Determine the exact average value of $f(x,y) = y$ over the upper half of $D$.

3. Find the exact center of mass of the lamina over the portion of $D$ that lies in the first quadrant and has its mass density distribution given by $\delta(x,y) = 1$. (Before making any calculations…

For each of the following iterated integrals,

• sketch and label the region of integration,

• convert the integral to the other coordinate system (if given in polar, to rectangular; if given in rectangular, to polar), and

• choose one of the two iterated integrals to evaluate exactly.

1. $\int_{\pi}^{3\pi/2} \int_{0}^{3} r^3 \, dr \, d\theta$

2. $\int_{0}^{2} \int_{-\sqrt{1-(x-1)^2}}^{\sqrt{1-(x-1)^2}} \sqrt{x^2 + y^2} \, dy \, dx$

3. $\int_0^{\pi/2} \int_0^{\sin(\theta)} r \sqrt{1-r^2} \, dr \, d\theta.$

4. $\int_0^{\sqrt{2}/2} \int_y^{\sqrt{1-y^2}} \cos(x^2 + y^2) \, dx \, dy.$