91Ó°ÊÓ

\(3-14\) Calculate the iterated integral. $$\int_{0}^{2} \int_{0}^{1}(2 x+y)^{8} d x d y$$

Find the image of the set \(S\) under the given transformation. \(S=\\{(u, v) | 0 \leqslant u \leqslant 3,0 \leqslant v \leqslant 2\\}\) \(x=2 u+3 v, y=u-v\)

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

\(3-10\) Find the mass and center of mass of the lamina that occupies the region \(D\) and has the given density function \(\rho .\) \(D\) is bounded by \(y=\sqrt{x}, y=0,\) and \(x=1 ; \rho(x, y)=x\)

\(7-8\) Identify the surface whose equation is given. $$\rho^{2}\left(\sin ^{2} \phi \sin ^{2} \theta+\cos ^{2} \phi\right)=9$$

\(3-14\) Calculate the iterated integral. $$\int_{0}^{1} \int_{1}^{2} \frac{x e^{x}}{y} d y d x$$

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

Evaluate the double integral. $$\iint_{D} \frac{y}{x^{5}+1} d A, \quad D=\\{(x, y) | 0 \leqslant x \leqslant 1,0 \leqslant y \leqslant x^{2}\\}$$

\(7-14\) Evaluate the given integral by changing to polar coordinates. \(\int_{R}(x+y) d A,\) where \(R\) is the region that lies to the left of the \(y\) -axis between the circles \(x^{2}+y^{2}=1\) and \(x^{2}+y^{2}=4\)

Find the image of the set \(S\) under the given transformation. \(S\) is the square bounded by the lines \(u=0, u=1, v=0\) \(v=1 ; \quad x=v, y=u\left(1+v^{2}\right)\)