91Ó°ÊÓ

In Problems 1–10, evaluate the iterated integrals. $$ \int_{-2}^{4} \int_{x-1}^{x+1} \int_{0}^{\sqrt{2 y / x}} 3 x y z d z d y d x $$

In Problems \(7-14\), use cylindrical coordinates to find the indicated quantity. Volume of the solid bounded above by the sphere centered at the origin having radius 5 and below by the plane \(z=4\).

In Problems 7-10, find the image of the rectangle with the given corners and find the Jacobian of the transformation. $$ x=u^{2}+v^{2}, y=v ;(0,0),(1,0),(1,1),(0,1) $$

Evaluate each of the iterated integrals. \(\int_{0}^{\pi / 2} \int_{0}^{1} x \sin x y d y d x\)

The part of the sphere \(x^{2}+y^{2}+z^{2}=a^{2}\) inside the elliptic cylinder \(b^{2} x^{2}+a^{2} y^{2}=a^{2} b^{2}\), where \(0

In Problems 1–10, evaluate the iterated integrals. $$ \int_{0}^{\pi / 2} \int_{\sin 2 z}^{0} \int_{0}^{2 y z} \sin \left(\frac{x}{y}\right) d x d y d z $$

In Problems 7-10, find the image of the rectangle with the given corners and find the Jacobian of the transformation. $$ x=u, y=u^{2}-v^{2} ;(0,0),(3,0),(3,1),(0,1) $$

\(S\) is the region inside the cardioid \(r=6-6 \sin \theta\).

In Problems 1-10, 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. \(r=2+2 \cos \theta ; \delta(r, \theta)=r\)

Evaluate the iterated integrals in Problems 1-14. \(\int_{0}^{2} \int_{-x}^{x} e^{-x^{2}} d y d x\)