91Ó°ÊÓ

Evaluate each of the iterated integrals. $$ \int_{0}^{\pi} \int_{0}^{1} x \sin y d x d y $$

Evaluate the iterated integrals. \(\int_{0}^{2} \int_{1}^{z} \int_{0}^{\sqrt{x / z}} 2 x y z d y d x d z\)

Evaluate the iterated integrals. \(\int_{0}^{\pi / 2} \int_{0}^{z} \int_{0}^{y} \sin (x+y+z) d x d y d z\)

Evaluate each of the iterated integrals. $$ \int_{0}^{\ln 3} \int_{0}^{\ln 2} e^{x+y} d y d x $$

Find the area of the given region \(S\) by calculating \(\iint_{S} r d r d \theta .\) Be sure to make a sketch of the region first. \(S\) is the smaller region bounded by \(\theta=\pi / 6\) and \(r=4 \sin \theta\).

Find the mass \(m\) and center of mass \((\bar{x}, \bar{y})\) of the lamina bounded by the given curves and with the indicated density. \(r=1+\cos \theta ; \delta(r, \theta)=r\)

Find the image of the rectangle with the given corners and find the Jacobian of the transformation. $$ x=2 u+3 v, y=u-v ;(0,0),(3,0),(3,1),(0,1) $$

Use cylindrical coordinates to find the indicated quantity. Volume of the solid bounded above by the sphere \(x^{2}+y^{2}+z^{2}=9\), below by the plane \(z=0\), and laterally by the cylinder \(x^{2}+y^{2}=4\)

Find the area of the indicated surface. Make a sketch in each case. The part of the surface \(z=x^{2} / 4+4\) that is cut off by the planes \(x=0, x=1, y=0\), and \(y=2\).

Evaluate the iterated integrals. $$ \int_{0}^{\pi / 4} \int_{\sqrt{2}}^{\sqrt{2} \cos \theta} r d r d \theta $$