91Ó°ÊÓ

Which do you think will be larger, the average value of \(f(x, y)=x y\) over the square \(0 \leq x \leq 1,0 \leq y \leq 1,\) or the average value of \(f\) over the quarter circle \(x^{2}+y^{2} \leq 1\) in the first quadrant? Calculate them to find out.

Evaluate the spherical coordinate integrals. $$\int_{0}^{\pi} \int_{0}^{\pi} \int_{0}^{2 \sin \phi} \rho^{2} \sin \phi d \rho d \phi d \theta$$

Change the Cartesian integral into an equivalent polar integral. Then evaluate the polar integral. $$\int_{0}^{1} \int_{x}^{\sqrt{2-x^{2}}}(x+2 y) d y d x$$

Evaluate the double integral over the given region \(R\). $$\iint_{R} \frac{x y^{3}}{x^{2}+1} d A, \quad R: \quad 0 \leq x \leq 1, \quad 0 \leq y \leq 2$$

Sketch the region of integration and evaluate the integral. $$\int_{1}^{\ln 8} \int_{0}^{\ln y} e^{x+y} d x d y$$

Find the Jacobian \(\partial(x, y) / \partial(u, v)\) of the transformation a. \(x=u \cos v, \quad y=u \sin v\) b. \(x=u \sin v, \quad y=u \cos v\)

Find the average height of the paraboloid \(z=x^{2}+y^{2}\) over the square \(0 \leq x \leq 2,0 \leq y \leq 2\).

Evaluate the spherical coordinate integrals. $$\int_{0}^{2 \pi} \int_{0}^{\pi / 4} \int_{0}^{2}(\rho \cos \phi) \rho^{2} \sin \phi d \rho d \phi d \theta$$

Find the Jacobian \(\partial(x, y, z) / \partial(u, v, w)\) of the transformation a. \(x=u \cos v, \quad y=u \sin v, \quad z=w\) b. \(x=2 u-1, \quad y=3 v-4, \quad z=(1 / 2)(w-4)\)

Change the Cartesian integral into an equivalent polar integral. Then evaluate the polar integral. $$\int_{1}^{2} \int_{0}^{\sqrt{2 x-x^{2}}} \frac{1}{\left(x^{2}+y^{2}\right)^{2}} d y d x$$