91Ó°ÊÓ

Find the following average values. The average of the squared distance between the origin and points in the solid paraboloid \(D=\left\\{(x, y, z): 0 \leq z \leq 4-x^{2}-y^{2}\right\\}\)

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

Evaluate the following integrals as they are written. $$\int_{0}^{\pi / 2} \int_{0}^{\cos y} e^{\sin y} d x d y$$

Find the volume of the following solids. The solid beneath the cylinder \(f(x, y)=e^{-x}\) and above the region \(R=\\{(x, y): 0 \leq x \leq \ln 4,-2 \leq y \leq 2\\}\)

Spherical coordinates Evaluate the Jacobian for the transformation from spherical to rectangular coordinates: \(x=\rho \sin \varphi \cos \theta, y=\rho \sin \varphi \sin \theta, z=\rho \cos \varphi .\) Show that \(J(\rho, \varphi, \theta)=\rho^{2} \sin \varphi\)

Find the following average values. The average \(z\) -coordinate of points on and within a hemisphere of radius 4 centered at the origin with its base in the \(x y\) -plane

Use spherical coordinates to find the volume of the following solids. The solid bounded by the sphere \(\rho=2 \cos \varphi\) and the hemisphere \(\rho=1, z \geq 0\)

Evaluate the following integrals. A sketch is helpful. \(\iint_{R} 12 y d A ; R\) is bounded by \(y=2-x, y=\sqrt{x},\) and \(y=0\).

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

Find the volume of the following solids. The solid beneath the plane \(f(x, y)=6-x-2 y\) and above the region \(R=\\{(x, y): 0 \leq x \leq 2,0 \leq y \leq 1\\}\)