91Ó°ÊÓ

Describe in words the surface whose equation is given. $$\theta=\pi / 4$$

Find the Jacobian of the transformation. $$x=u / v, \quad y=v / w, \quad z=w / u$$

Sketch the region whose area is given by the integral and evaluate the integral. \(\int_{\pi / 2}^{\pi} \int_{0}^{2 \sin \theta} r d r d \theta\)

\(D\) is the triangular region enclosed by the lines \(x=0\) \(y=x,\) and \(2 x+y=6 ; \rho(x, y)=x^{2}\)

\(\mathrm{A} 20-\mathrm{ft}-\mathrm{by}-30-\mathrm{ft}\)swimming pool is filled with water. The depth is measured at 5 -foot intervals, starting at one corner of the pool, and the values are recorded in the table. Estimate the volume of water in the pool. $$\begin{array}{|c|c|c|c|c|c|c|c|}\hline & {0} & {5} & {10} & {15} & {20} & {25} & {30} \\ \hline 0 & {2} & {3} & {4} & {6} & {7} & {8} & {8} \\ {5} & {2} & {3} & {4} & {7} & {8} & {10} & {8} \\ {10} & {2} & {4} & {6} & {8} & {10} & {12} & {10} \\ {15} & {2} & {3} & {4} & {5} & {6} & {8} & {7} \\ {20} & {2} & {2} & {2} & {2} & {3} & {4} & {4} \\ \hline\end{array}$$

Evaluate the iterated integral. $$\int_{0}^{\sqrt{\pi}} \int_{0}^{x} \int_{0}^{x z} x^{2} \sin y d y d z d x$$

Find the Jacobian of the transformation. $$x=v+w^{2}, \quad y=w+u^{2}, \quad z=u+v^{2}$$

\(5-6\) Describe in words the surface whose equation is given. $$\rho=3$$

Describe in words the surface whose equation is given. $$r=5$$

\(1-6=\) Evaluate the iterated integral. \(\int_{0}^{1} \int_{0}^{e^{v}} \sqrt{1+e^{v}} d w d v\)