91Ó°ÊÓ

Sketch the region of integration and evaluate the following integrals as they are written. $$\int_{0}^{\pi / 2} \int_{y}^{\pi / 2} 6 \sin (2 x-3 y) d x d y$$

Evaluate the following iterated integrals. $$\int_{0}^{1} \int_{1}^{4} \frac{3 y}{\sqrt{x+y^{2}}} d x d y$$

Evaluate the Jacobian for the transformation from cylindrical coordinates \((r, \theta, Z)\) to rectangular coordinates \((x, y, z): x=r \cos \theta, y=r \sin \theta, z=Z .\) Show that \(J(r, \theta, Z)=r\)

Use spherical coordinates to find the volume of the following regions. A ball of radius \(a>0\)

Evaluate the following iterated integrals. $$\int_{1}^{4} \int_{0}^{2} e^{\sqrt{x}} d y d x$$

Find the following average values. The average distance between points within the cardioid \(r=1+\cos \theta\) and the origin

Use polar coordinates to find the centroid of the following constant-density plane regions. The quarter-circular disk \(R=\\{(r, \theta): 0 \leq r \leq 2\) \(0 \leq \theta \leq \pi / 2\\}\)

Sketch the region of integration and evaluate the following integrals as they are written. $$\int_{0}^{\pi / 2} \int_{0}^{\cos y} e^{\sin y} d x d y$$

Use another order of integration to evaluate \(\int_{1}^{4} \int_{z}^{4 z} \int_{0}^{\pi^{2}} \frac{\sin \sqrt{y z}}{x^{3 / 2}} d y d x d z\)

Use polar coordinates to find the centroid of the following constant-density plane regions. The region bounded by the cardioid \(r=1+\cos \theta\)