91Ó°ÊÓ

Evaluate the integral. $$\int_{0}^{x}(2 x-y) d y$$

Find the mass of the lamina described by the inequalities, given that its density is \(\rho(x, y)=x y .\) (Hint: Some of the integrals are simpler in polar coordinates.) \(0 \leq x \leq 4,0 \leq y \leq 3\)

Find the area of the surface given by \(z=f(x, y)\) over the region \(R .\) (Hint: Some of the integrals are simpler in polar coordinates.) $$ \begin{aligned} &f(x, y)=2 x+2 y\\\ &R \text { : triangle with vertices }(0,0),(2,0),(0,2) \end{aligned} $$

Find the Jacobian \(\partial(x, y) / \partial(u, v)\) for the indicated change of variables. $$ x=-\frac{1}{2}(u-v), y=\frac{1}{2}(u+v) $$

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

Evaluate the iterated integral. \(\int_{0}^{3} \int_{0}^{2} \int_{0}^{1}(x+y+z) d x d y d z\)

Evaluate the iterated integral. $$ \int_{0}^{\pi / 4} \int_{0}^{2} \int_{0}^{2-r} r z d z d r d \theta $$

Evaluate the iterated integral. \(\int_{-1}^{1} \int_{-1}^{1} \int_{-1}^{1} x^{2} y^{2} z^{2} d x d y d z\)

Find the Jacobian \(\partial(x, y) / \partial(u, v)\) for the indicated change of variables. $$ x=a u+b v, y=c u+d v $$

Find the area of the surface given by \(z=f(x, y)\) over the region \(R .\) (Hint: Some of the integrals are simpler in polar coordinates.) $$ \begin{aligned} &f(x, y)=15+2 x-3 y\\\ &R \text { : square with vertices }(0,0),(3,0),(0,3),(3,3) \end{aligned} $$