91Ó°ÊÓ

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

Electric charge is distributed over the disk \(x^{2}+y^{2} \leqslant 1\) so that the charge density at \((x, y)\) is \(\sigma(x, y)=\sqrt{x^{2}+y^{2}}\) (measured in coulombs per square meter). Find the total charge on the disk.

Find the Jacobian of the transformation. $$x=u v, \quad y=u / v$$

Find the mass and center of mass of the lamina that occupies the region \(D\) and has the given density function \(\rho\) . $$D=\\{(x, y) | 1 \leqslant x \leqslant 3,1 \leqslant y \leqslant 4\\} ; \rho(x, y)=k y^{2}$$

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

\(1-6=\) Evaluate the iterated integral. $$\int_{0}^{1} \int_{x^{2}}^{x}(1+2 y) d y d x$$

Find the Jacobian of the transformation. $$x=e^{-r} \sin \theta, \quad y=e^{r} \cos \theta$$

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

(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).

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