In this exercise, use polar coordinates to describe the region shown.

In this exercise, use polar coordinates to describe the region shown.

A region R is shown. Decide whether to use polar coordinates or rectangular coordinates and write $\iint_R f(x, y) d A$ as an iterated integral, where $f$ is an arbitrary continuous function on $R$.

A region $R$ is shown. Decide whether to use polar coordinates or rectangular coordinates and write $\iint_{R}f(x, y)~dA$ as an iterated integral, where $f$ is an arbitrary continuous function on $R$.

A region $R$ is given. Decide whether to use polar coordinates or rectangular coordinates and write $\iint_{R}f(x, y)~dA$ as an iterated integral, where $f$ is an arbitrary continuous function on $R$.

Find the area of the region that lies outside the circle $x^2+y^2=4$ but inside the circle

$$x^2+y^2-4 y-12=0$$

Identity the type of ODE and then solve.

1. dy/dx = (x2y – y)/(y+1) y(0) = 1
2. dy/dx = (x2 – y)/x
3. (x2 + y2 +x) dx + xy dy = 0
4. dy/dx = y/x + y2/x2 with y(1) = 1

Change $(\sqrt{3}, 1,2 \sqrt{3})$ from rectangular to spherical coordinates.

Convert from rectangular to spherical coordinates.

$$( \sqrt { 3 } , 1,2 \sqrt { 3 } )$$

Use cylindrical coordinates. Evaluate triple integral E z dV, where E is enclosed by the paraboloid z=x^2+y^2 and the plane z=4

Evaluate the given iterated integral. $\int_{0}^{1} \int_{0}^{1} \int_{0}^{2-x^{2}-y^{2}} x y e^{z} d z d x d y$

Evaluate the iterated integral. ∫_(-1)^1∫_(-2)^2 (x² - y²) dy dx

Find the period, and sketch the graph. $y=\sec \left(\frac{1}{2} x-\frac{\pi}{2}\right)$

Use a geometric argument to show that ∫_0^3∫_0^9-y² √9-x²-y² dx dy = 9π/2.