Problem 17
Evaluate the triple integral. $$ \int_{-\pi / 2}^{\pi / 2} \int_{0}^{\pi} \int_{0}^{\pi}(\cos x \sin y \sin z) d z d y d x $$
Problem 18
Find the mass/weight of the lamina described by the region \(R\) in the plane and its density function \(\delta(x, y)\). \(R\) is the annulus in the first and second quadrants bounded by \(x^{2}+y^{2}=9\) and \(x^{2}+y^{2}=36 ; \delta(x, y)=\sqrt{x^{2}+y^{2}} \mid b / f t^{2}\)
Problem 18
Iterated integrals are given that compute the area of a region \(R\) in the \(x\) -y plane. Sketch the region \(R\), and give the iterated integral(s) that give the area of \(R\) with the opposite order of integration. $$ \int_{0}^{1} \int_{5-5 x}^{5-5 x^{2}} d y d x $$
Problem 18
Find the area of the given surface over the region \(R\). Find the surface area of the sphere with radius 5 by doubling the surface area of \(f(x, y)=\sqrt{25-x^{2}-y^{2}}\) over \(R\), bounded by the circle \(x^{2}+y^{2}=25 .\) (Be sure to compare your result with the known formula.)
Problem 18
(a) Sketch the region \(R\) given by the problem. (b) Set up the iterated integrals, in both orders, that evaluate the given double integral for the described region \(R\) (c) Evaluate one of the iterated integrals to find the signed volume under the surface \(z=f(x, y)\) over the region \(R .\) \(\iint_{R}(4-3 y) d A,\) where \(R\) is bounded by \(y=0, y=x / e\) and \(y=\ln x\).
Problem 18
A triple integral in spherical coordinates is given. Describe the region in space defined by the bounds of the integral. $$ \int_{0}^{\pi} \int_{0}^{\pi} \int_{1}^{1.1} \rho^{2} \sin (\varphi) d \rho d \theta d \varphi $$
Problem 18
Evaluate the triple integral. $$ \int_{0}^{1} \int_{0}^{x} \int_{0}^{x+y}(x+y+z) d z d y d x $$
Problem 19
A triple integral in spherical coordinates is given. Describe the region in space defined by the bounds of the integral. $$ \int_{0}^{2 \pi} \int_{0}^{\pi / 4} \int_{0}^{2} \rho^{2} \sin (\varphi) d \rho d \theta d \varphi $$
Problem 19
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. $$ \int_{0}^{4} \int_{y / 2}^{2} e^{x^{2}} d x d y $$
Problem 19
In Exercises \(19-26,\) 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)\(; \delta(x, y)=5 \mathrm{gm} / \mathrm{cm}^{2}\)
