Problem 5
In Exercises \(5-10,\) evaluate the double integral \(\int_{R} \int f(r, \theta) d A,\) and sketch the region \(R\). $$ \int_{0}^{2 \pi} \int_{0}^{6} 3 r^{2} \sin \theta d r d \theta $$
Problem 5
In Exercises \(1-8,\) find the Jacobian \(\partial(x, y) / \partial(u, v)\) for the indicated change of variables. \(x=u \cos \theta-v \sin \theta, y=u \sin \theta+v \cos \theta\)
Problem 5
Evaluate the iterated integral. $$ \int_{0}^{4} \int_{0}^{\pi / 2} \int_{0}^{1-x} x \cos y d z d y d x $$
Problem 5
In Exercises \(1-10\), evaluate the integral. $$ \int_{0}^{\sqrt{4-x^{2}}} x^{2} y d y $$
Problem 6
Evaluate the iterated integral. $$ \int_{0}^{\pi / 2} \int_{0}^{y / 2} \int_{0}^{1 / y} \sin y d z d x d y $$
Problem 6
Sketch the region \(R\) and evaluate the iterated integral \(\int_{R} \int f(x, y) d A .\) $$ \int_{0}^{\pi} \int_{0}^{\pi / 2} \sin ^{2} x \cos ^{2} y d y d x $$
Problem 6
In Exercises \(1-8,\) find the Jacobian \(\partial(x, y) / \partial(u, v)\) for the indicated change of variables. \(x=u+a, y=v+a\)
Problem 6
Evaluate the double integral \(\int_{R} \int f(r, \theta) d A,\) and sketch the region \(R\). $$ \int_{0}^{\pi / 4} \int_{0}^{4} r^{2} \sin \theta \cos \theta d r d \theta $$
Problem 6
Find the area of the surface given by \(z=f(x, y)\) over the region \(R .\) $$ \begin{aligned} &f(x, y)=9+x^{2}-y^{2}\\\ &R=\left\\{(x, y): x^{2}+y^{2} \leq 4\right\\} \end{aligned} $$
Problem 6
Find the mass and center of mass of the lamina for each density. \(R:\) triangle with vertices \((0,0),(0, a),(a, 0)\) (a) \(\rho=k\) (b) \(\rho=x^{2}+y^{2}\)
