Problem 8
Evaluate the double integral \(\int_{R} \int f(r, \theta) d A,\) and sketch the region \(R\). $$ \int_{0}^{\pi / 2} \int_{0}^{3} r e^{-r^{2}} d r d \theta $$
Problem 8
Sketch the region \(R\) and evaluate the iterated integral \(\int_{R} \int f(x, y) d A .\) $$ \int_{0}^{4} \int_{\frac{1}{2} y}^{\sqrt{y}} x^{2} y^{2} d x d y $$
Problem 8
In Exercises \(1-10\), evaluate the integral. $$ \int_{-\sqrt{1-y^{2}}}^{\sqrt{1-y^{2}}}\left(x^{2}+y^{2}\right) d x $$
Problem 8
In Exercises \(1-8,\) find the Jacobian \(\partial(x, y) / \partial(u, v)\) for the indicated change of variables. \(x=\frac{u}{v}, y=u+v\)
Problem 9
In Exercises 9-12, use cylindrical coordinates to find the volume of the solid. Solid inside both \(x^{2}+y^{2}+z^{2}=a^{2}\) and \((x-a / 2)^{2}+y^{2}=(a / 2)^{2}\)
Problem 9
Evaluate the double integral \(\int_{R} \int f(r, \theta) d A,\) and sketch the region \(R\). $$ \int_{0}^{\pi / 2} \int_{0}^{1+\sin \theta} \theta r d r d \theta $$
Problem 9
Set up a triple integral for the volume of the solid. The solid bounded by the paraboloid \(z=9-x^{2}-y^{2}\) and the plane \(z=0\)
Problem 9
Sketch the region \(R\) and evaluate the iterated integral \(\int_{R} \int f(x, y) d A .\) $$ \int_{-a}^{a} \int_{-\sqrt{a^{2}-x^{2}}}^{\sqrt{a^{2}-x^{2}}}(x+y) d y d x $$
Problem 9
In Exercises \(1-10\), evaluate the integral. $$ \int_{0}^{x^{3}} y e^{-y / x} d y $$
Problem 9
Find the area of the surface. The portion of the plane \(z=24-3 x-2 y\) in the first octant
