91Ó°ÊÓ

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 the iterated integrals. $$ \int_{0}^{\pi / 9} \int_{\pi / 4}^{3 r} \sec ^{2} \theta d \theta d r $$

Find the area of the indicated surface. Make a sketch in each case. The part of the sphere \(x^{2}+y^{2}+z^{2}=a^{2}\) inside the circular cylinder \(x^{2}+y^{2}=b^{2}\), where \(0

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. \(r=1, r=2, \theta=0, \theta=\pi,(0 \leq \theta \leq \pi) ; \delta(r, \theta)=1 / r\)

Evaluate the iterated integrals. \(\int_{-2}^{4} \int_{x-1}^{x+1} \int_{0}^{\sqrt{2 y / x}} 3 x y z d z d y d x\)

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 \(z=4\).

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

Find the area of the indicated surface. Make a sketch in each case. 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

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

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