91Ó°ÊÓ

Find a parametric representation of the surface in terms of the parameters \(u=x\) and \(v=y\). (a) \(z+z x^{2}-y=0\) (b) \(y^{2}-3 z=5\)

Use spherical coordinates to find the volume of the solid. The solid within the cone \(\phi=\pi / 4\) and between the spheres \(\rho=1\) and \(\rho=2\).

$$ \begin{aligned} &\iint_{R} \frac{x y}{\sqrt{x^{2}+y^{2}+1}} d A \\ &R=\\{(x, y): 0 \leq x \leq 1,0 \leq y \leq 1\\} \end{aligned} $$

Use the numerical triple integral operation of a CAS to approximate $$\iiint_{G} e^{-x^{2}-y^{2}-z^{2}} d V$$ where \(G\) is the spherical region \(x^{2}+y^{2}+z^{2} \leq 1\)

Determine whether the statement is true or false. Explain your answer. If \(\mathbf{r}=x(u, v) \mathbf{i}+y(u, v) \mathbf{j}\) maps the rectangle \(0 \leq u \leq 2\), \(1 \leq v \leq 5\) to a region \(R\) in the \(x y\) -plane, then the area of \(R\) is given by $$\int_{1}^{5} \int_{0}^{2}\left|\frac{\partial(x, y)}{\partial(u, v)}\right| d u d v$$

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. \(\iint_{R} x^{2} d A ; R\) is the region bounded by \(y=16 / x, y=x\), and \(x=8\).

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 $$\frac{\partial(x, y)}{\partial(r, \theta)}=r^{2}$$

Use spherical coordinates to find the volume of the solid. The solid enclosed by the sphere \(x^{2}+y^{2}+z^{2}=4 a^{2}\) and the planes \(z=0\) and \(z=a\).

(a) Find parametric equations for the portion of the cylin\(\operatorname{der} x^{2}+y^{2}=5\) that extends between the planes \(z=0\) and \(z=1\). (b) Find parametric equations for the portion of the cylin\(\operatorname{der} x^{2}+z^{2}=4\) that extends between the planes \(y=1\) and \(y=3\).

$$ \iint_{R} x \sqrt{1-x^{2}} d A ; R=\\{(x, y): 0 \leq x \leq 1,2 \leq y \leq 3\\} $$