Problem 19
Evaluate the triple integral. $$ \int_{0}^{\pi} \int_{0}^{1} \int_{0}^{z}(\sin (y z)) d x d y d z $$
Problem 20
State why it is difficult/impossible to integrate the iterated integral in the given order of integration. Change the order of integration and evaluate the new iterated integral. $$ \int_{0}^{\sqrt{\pi / 2}} \int_{x}^{\sqrt{\pi / 2}} \cos \left(y^{2}\right) d y d x $$
Problem 20
Evaluate the triple integral. $$ \int_{\pi}^{\pi^{2}} \int_{x}^{x^{3}} \int_{-y^{2}}^{y^{2}}\left(z \frac{x^{2} y+y^{2} x}{e^{x^{2}+y^{2}}}\right) d z d y d x $$
Problem 20
A triple integral in spherical coordinates is given. Describe the region in space defined by the bounds of the integral. $$ \int_{0}^{2 \pi} \int_{\pi / 6}^{\pi / 4} \int_{0}^{2} \rho^{2} \sin (\varphi) d \rho d \theta d \varphi $$
Problem 20
Find the center of mass of the lamina described by the region \(R\) in the plane and its density function \(\delta(x, y)\) Note: these are the same lamina as in Exercises \(11-18\). \(R\) is the rectangle with corners (1,-3),(1,2),(7,2) and (7,-3)\(; \delta(x, y)=\left(x+y^{2}\right) \mathrm{gm} / \mathrm{cm}^{2}\)
Problem 21
State why it is difficult/impossible to integrate the iterated integral in the given order of integration. Change the order of integration and evaluate the new iterated integral. $$ \int_{0}^{1} \int_{y}^{1} \frac{2 y}{x^{2}+y^{2}} d x d y $$
Problem 21
A triple integral in spherical coordinates is given. Describe the region in space defined by the bounds of the integral. $$ \int_{0}^{2 \pi} \int_{0}^{\pi / 6} \int_{0}^{\sec \varphi} \rho^{2} \sin (\varphi) d \rho d \theta d \varphi $$
Problem 21
Find the center of mass of the lamina described by the region \(R\) in the plane and its density function \(\delta(x, y)\) Note: these are the same lamina as in Exercises \(11-18\). \(R\) is the triangle with corners \((-1,0),(1,0),\) and (0,1) \(\delta(x, y)=2 \mathrm{lb} / \mathrm{in}^{2}\)
Problem 21
Find the center of mass of the solid represented by the indicated space region \(D\) with density function \(\delta(x, y, z)\). \(D\) is bounded by the coordinate planes and \(z=2-2 x / 3-2 y ; \quad \delta(x, y, z)=10 \mathrm{gm} / \mathrm{cm}^{3}\).
Problem 22
State why it is difficult/impossible to integrate the iterated integral in the given order of integration. Change the order of integration and evaluate the new iterated integral. $$ \int_{-1}^{1} \int_{1}^{2} \frac{x \tan ^{2} y}{1+\ln y} d y d x $$
