91Ó°ÊÓ

Plot the point whose cylindrical coordinates are given. Then find the rectangular coordinates of the point. $$ \text { (a) }(\sqrt{2}, 3 \pi / 4,2) \quad \text { (b) }(1,1,1) $$

If \(R=[0,4] \times[-1,2]\), use a Riemann sum with \(m=2\) \(n=3\) to estimate the value of \(\iint_{R}\left(1-x y^{2}\right) d A .\) Take the sample points to be (a) the lower right corners and (b) the upper left corners of the rectangles.

Plot the point whose spherical coordinates are given. Then find the rectangular coordinates of the point. $$ \text { (a) }(2, \pi / 2, \pi / 2) \quad \text { (b) } \quad(4,-\pi / 4, \pi / 3) $$

Find the Jacobian of the transformation. $$ x=u^{2}+u v, \quad y=u v^{2} $$

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

Find the area of the surface. The part of the plane \(3 x+2 y+z=6\) that lies in the first octant.

\(3-4\) Change from rectangular to cylindrical coordinates. $$ \text { (a) }(-1,1,1) \quad \text { (b) }(-2,2 \sqrt{3}, 3) $$

\(3-4\) Change from rectangular to spherical coordinates. $$ \begin{array}{lll}{\text { (a) }(0,-2,0)} & {\text { (b) }(-1,1,-\sqrt{2})} & {}\end{array} $$

Evaluate the iterated integral. $$ \int_{0}^{1} \int_{0}^{y} x e^{y^{3}} d x d y $$

(a) Use a Riemann sum with \(m=n=2\) to estimate the value of \(\iint_{R} x e^{-x y} d A,\) where \(R=[0,2] \times[0,1]\). Take the sample points to be upper right corners. (b) Use the Midpoint Rule to estimate the integral in part (a).