91Ó°ÊÓ

\(7-10=\) Evaluate the double integral. $$\iint_{D} x d A, \quad D=\\{(x, y) | 0 \leqslant x \leqslant \pi, 0 \leqslant y \leqslant \sin x\\}$$

Find the image of the set \(S\) under the given transformation. $$\begin{array}{l}{S \text { is the triangular region with vertices }(0,0),(1,1),(0,1) ;} \\ {x=u^{2}, y=v}\end{array}$$

\(9-10\) Write the equation in spherical coordinates. $$\text {(a)}x^{2}-2 x+y^{2}+z^{2}=0 \quad \text { (b) } x+2 y+3 z=1$$

The integral \(\iint_{R} \sqrt{9-y^{2}} d A,\) where \(R=[0,4] \times[0,2]\) represents the volume of a solid. Sketch the solid.

Write the equations in cylindrical coordinates. $$(a)3 x+2 y+z=6 \quad \text { (b) }-x^{2}-y^{2}+z^{2}=1$$

\(\iiint_{L} \sin y d V,\) where \(E\) lies below the plane \(z=x\) and above the triangular region with vertices \((0,0,0),\) \((\pi, 0,0),\) and \((0, \pi, 0)\)

Find the image of the set \(S\) under the given transformation. $$S \text { is the disk given by } u^{2}+v^{2} \leqslant 1 ; \quad x=a u, y=b v$$

Evaluate the given integral by changing to polar coordinates. \(\iint_{R} \frac{y^{2}}{x^{2}+y^{2}} d A,\) where \(R\) is the region that lies between the circles \(x^{2}+y^{2}=a^{2}\) and \(x^{2}+y^{2}=b^{2}\) with \(0 < a < b\)

\(7-10=\) Evaluate the double integral. $$\iint_{D} x^{3} d A, \quad D=\\{(x, y) | 1 \leqslant x \leqslant e, 0 \leqslant y \leqslant \ln x\\}$$

Find the mass and center of mass of the lamina that occupies the region and has the given density function . $$\begin{array}{l}{D \text { is bounded by the parabolas } y=x^{2} \text { and } x=y^{2}} \\ {\rho(x, y)=\sqrt{x}}\end{array}$$