91Ó°ÊÓ

Determine whether the statement is true or false. Explain your answer. The Jacobian of the transformation \(x=r \cos \theta, y=r \sin \theta\) is $$\frac{\partial(x, y)}{\partial(r, \theta)}=r^{2}$$

(a) Find parametric equations for the portion of the plane \(x+y=1\) that extends between the planes \(z=-1\) and \(z=1\) (b) Find parametric equations for the portion of the plane \(y-2 z=5\) that extends between the planes \(x=0\) and \(x=3\)

Use spherical coordinates to find the volume of the solid. The solid within the sphere \(x^{2}+y^{2}+z^{2}=9\), outside the cone \(z=\sqrt{x^{2}+y^{2}}\), and above the \(x y\) -plane.

Determine whether the statement is true or false. Explain your answer. The Jacobian of the transformation \(x=\rho \sin \phi \cos \theta\) \(y=\rho \sin \phi \sin \theta, z=\rho \cos \phi\) is $$\frac{\partial(x, y, z)}{\partial(\rho, \phi, \theta)}=\rho^{2} \sin \phi$$

$$ \begin{aligned} &\iint_{R}(x \sin y-y \sin x) d A \\ &R=\\{(x, y): 0 \leq x \leq \pi / 2,0 \leq y \leq \pi / 3\\} \end{aligned} $$

Use a triple integral to find the volume of the solid. The solid bounded by the surface \(z=\sqrt{y}\) and the planes \(x+y=1, x=0\), and \(z=0\)

Evaluate the double integral in two ways using iterated integrals: (a) viewing \(R\) as a type I region, and (b) viewing \(R\) as a type II region. \(\iint_{R} x y^{2} d A ; R\) is the region enclosed by \(y=1, y=2\), \(x=0\), and \(y=x\).

Use cylindrical or spherical coordinates to evaluate the integral. \(\int_{0}^{a} \int_{0}^{\sqrt{a^{2}-x^{2}}} \int_{0}^{a^{2}-x^{2}-y^{2}} x^{2} d z d y d x \quad(a>0)\)

Use a triple integral to find the volume of the solid. The solid bounded by the surface \(y=x^{2}\) and the planes \(y+z=4\) and \(z=0\)

Find parametric equations for the surface generated by revolving the curve \(y=\sin x\) about the \(x\) -axis.