Chapter 9

Show that the given integral is independent of the path. Evaluate. $$ \int_{(1,0,0)}^{(2, \pi / 2,1)}\left(2 x \sin y+e^{3 z}\right) d x+x^{2} \cos y d y+\left(3 x e^{3 z}+5\right) d z $$

Let \(\mathbf{a}\) be a constant voctor and \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\). Verify the given identity. $$ \nabla \cdot(\mathbf{a} \times \mathbf{r})=0 $$

Let \(C\) be a plane curve traced by \(r(t)=f(t) \dot{i}+g(t) \dot{j}\), where \(f\) and \(g\) have second derivatives. Show that the curvature at a point is given by $$ \kappa=\begin{aligned} &\left|f^{\prime}(t) g^{\prime \prime}(t)-g^{\prime}(t) f^{\prime \prime}(t)\right| \\ &\left(\left[f^{\prime}(t)\right]^{2}+\left[g^{\prime}(t)\right]^{2}\right)^{3 / 2} \end{aligned} $$

Graph the curve \(C\) that is described by \(\mathbf{r}\) and graph \(\mathbf{r}^{\prime}\) at the indicated value of \(\boldsymbol{t}\). $$ \mathbf{r}(t)=2 \cos t \mathbf{i}+6 \sin t \mathbf{j} ; t=\pi / 6 $$

Evaluate the given integral by means of the indicated change of variables. \(\iiint_{D}(4 z+2 x-2 y) d V\), where \(D\) is the parallelepiped \(1 \leq y+z \leq 3,-1 \leq-y+z \leq 1,0 \leq x-y \leq 3 ; u=y+z\) \(v=-y+z, w=x-y\)

In Problems, find the polar moment of inertia $$ I_{0}=\iint_{R} r^{2} \rho(r, \theta) d A=I_{x}+I_{y} $$ of the lamina that has the given shape and density. \(r=\theta, 0 \leq \theta \leq \pi, y=0 ;\) density at a point \(P\) proportional to the distance from the pole

Find the volume of the solid bounded by the graphs of the given equations. \(x^{2}+y^{2}=4, z=x+y\), the coordinate planes, first octant

Evaluate the surface integral \(\iint_{S} G(x, y, z) d S\). \(G(x, y, z)=2 z ; S\) that portion ofthe paraboloid \(2 z=1+x^{2}+y^{2}\) in the first octant bounded by \(x=0, y=\sqrt{3} x, z=1\)

Evaluate the double integral over the region \(R\) that is bounded by the graphs of the given equations. Choose the most convenient order of integration. $$ \iint_{R} x d A ; y=\tan ^{-1} x, y=0, x=1 $$

In Problems, show that the given integral is independent of the path. Evaluate. $$ \int_{(1,2,1)}^{(3,4,1)}(2 x+1) d x+3 y^{2} d y+\frac{1}{z} d z $$

Let a be a constant vector and \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\). Verify the given identity. $$ \mathbf{a} \times(\boldsymbol{\nabla} \times \mathbf{r})=\mathbf{0} $$

Find an equation of the tangent plane to the graph of the given equation at the indicated point. $$ x^{2} y^{3}+6 z=10 ;(2,1,1) $$

Find the first partial derivatives of the given function. $$ z=e^{x^{2} \tan ^{-1} y^{2}} $$

Find the polar moment of inertia $$ I_{0}=\iint_{R} r^{2} \rho(r, \theta) d A=I_{x}+I_{y} $$ of the lamina that has the given shape and density. \(r=\theta, 0 \leq \theta \leq \pi, y=0\); density at a point \(P\) proportional to the distance from the pole

Show that the given integral is independent of the path. Evaluate. $$ \int_{(1,2,1)}^{(3,4,1)}(2 x+1) d x+3 y^{2} d y+\frac{1}{z} d z $$

Let \(\mathbf{a}\) be a constant voctor and \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\). Verify the given identity. $$ \mathbf{a} \times(\nabla \times \mathbf{r})=\mathbf{0} $$

Graph the curve \(C\) that is described by \(\mathbf{r}\) and graph \(\mathbf{r}^{\prime}\) at the indicated value of \(\boldsymbol{t}\). $$ \mathbf{r}(t)=t^{3} \mathbf{i}+t^{2} \mathbf{j} ; t=-1 $$

In Problems 23-26, evaluate the given double integral by means of an appropriate change of variables. \(\int_{0}^{1} \int_{0}^{1-x} e^{(y-x) /(y+x)} d y d x \quad\) 24. \(\int_{-2}^{0} \int_{0}^{x+2} e^{y^{2}-2 x y+x^{2}} d y d x\)

Find the volume of the solid bounded by the graphs of the given equations. \(y=x^{2}+z^{2}, \quad y=8-x^{2}-z^{2}\)

Evaluate the surface integral \(\iint_{S} G(x, y, z) d S\). \(G(x, y, z)=24 \sqrt{y z} ; S\) that portion of the cylinder \(y=x^{2}\) in the first octant bounded by \(y=0, y=4, z=0, z=3\)

Evaluate \(\oint_{C}\left(x^{2}-y^{2}\right) d s\), where \(C\) is given by \(x=5 \cos t, \quad y=5 \sin t, \quad 0 \leq t \leq 2 \pi\).

Let a be a constant vector and \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\). Verify the given identity. $$ \nabla \times[(\mathbf{r} \cdot \mathbf{r}) \mathbf{a}]=2(\mathbf{r} \times \mathbf{a}) $$

Find the first partial derivatives of the given function. $$ f(x, y)=x e^{x^{3} y} $$

Suppose that \(\mathbf{r}(t)=r_{0} \cos \omega t \mathbf{i}+r_{0} \sin \omega t \mathbf{j}\) is the position vector of an object that is moving in a circle of radius \(r_{0}\) in the \(x y\) -plane. If \(\|\mathbf{v}(t)\|=v\), show that the magnitude of the centripetal acceleration is \(a=\|\mathbf{a}(t)\|=v^{2} / r_{0}\).

In Problems, find a vector that gives the direction in which the given function increases most rapidly at the indicated point. Find the maximum rate. $$ f(x, y)=e^{2 x} \sin y ;(0, \pi / 4) $$

Evaluate the given double integral by means of an appropriate change of variables. $$ \int_{0}^{1} \int_{0}^{1-x} e^{(y-x) /(y+x)} d y d x $$

Find the polar moment of inertia $$ I_{0}=\iint_{R} r^{2} \rho(r, \theta) d A=I_{x}+I_{y} $$ of the lamina that has the given shape and density. \(r \theta=1, \frac{1}{3} \leq \theta \leq 1, r=1, r=3, y=0\); density at a point \(P\) inversely proportional to the distance from the pole

Let \(\mathbf{a}\) be a constant voctor and \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\). Verify the given identity. $$ \nabla \times[(\mathbf{r} \cdot \mathbf{r}) \mathbf{a}]=2(\mathbf{r} \times \mathbf{a}) $$

Find an equation of the tangent plane to the graph of the given equation at the indicated point. $$ z=\ln \left(x^{2}+y^{2}\right) ;(1 / \sqrt{2}, 1 / \sqrt{2}, 0) $$

Find a vector that gives the direction in which the given function increases most rapidly at the indicated point. Find the maximum rate. $$ f(x, y)=e^{2 x} \sin y ;(0, \pi / 4) $$

Graph the curve \(C\) that is described by \(\mathbf{r}\) and graph \(\mathbf{r}^{\prime}\) at the indicated value of \(\boldsymbol{t}\). $$ \mathbf{r}(t)=2 \mathbf{i}+t \mathbf{j}+\frac{4}{1+t^{2}} \mathbf{k} ; t=1 $$

In Problems 23-26, evaluate the given double integral by means of an appropriate change of variables. \(\int_{-2}^{0} \int_{0}^{x+2} e^{y^{2}-2 x y+x^{2}} d y d x\)

Find the volume of the solid bounded by the graphs of the given equations. \(x=2, y=x, y=0, \quad z=x^{2}+y^{2}, \quad z=0\)

Evaluate the surface integral \(\iint_{S} G(x, y, z) d S\). \(G(x, y, z)=\left(1+4 y^{2}+4 z^{2}\right)^{1 / 2} ; S\) that portion of the paraboloid \(x=4-y^{2}-z^{2}\) in the first octant outside the cylinder \(y^{2}+z^{2}=1\)

In Problems, show that the given integral is independent of the path. Evaluate. $$ \int_{(-2,3,1)}^{(0,0,0)} 2 x z d x+2 y z d y+\left(x^{2}+y^{2}\right) d z $$

Evaluate \(\int_{-c} y d x-x d y\), where \(C\) is given by \(x=2 \cos t, \quad y=3 \sin t, \quad 0 \leq t \leq \pi\).

Let a be a constant vector and \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\). Verify the given identity. $$ \nabla \cdot[(\mathbf{r} \cdot \mathbf{r}) \mathbf{a}]=2(\mathbf{r} \cdot \mathbf{a}) $$

Verify that the given function satisfies Laplace's equation: $$\frac{\partial^{2} z}{\partial x^{2}}+\frac{\partial^{2} z}{\partial y^{2}}=0$$ $$ z=e^{x^{2}-y^{2}} \cos 2 x y $$

Find an equation of the tangent plane to the graph of the given equation at the indicated point. $$ z=8 e^{-2 y} \sin 4 x ;(\pi / 24,0,4) $$

Find the first partial derivatives of the given function. $$ f(\theta, \phi)=\phi^{2} \sin \frac{\theta}{\phi} $$

The motion of a particle in space is described by $$ \mathbf{r}(t)=b \cos t \mathbf{i}+b \sin t \mathbf{j}+c t \mathbf{k}, \quad t \geq 0. $$ (a) Compute \(\|\mathbf{v}(t)\|\). (b) Compute \(s=\int_{0}^{t}\|\mathbf{v}(t)\| d t\) and verify that \(d s / d t\) is the same as the result of part (a). (c) Verify that \(d^{2} s / d t^{2} \neq\|\mathbf{a}(t)\|\).

In Problems, find a vector that gives the direction in which the given function increases most rapidly at the indicated point. Find the maximum rate. $$ f(x, y)=x y e^{x-y} ;(5,5) $$

Evaluate the given double integral by means of an appropriate change of variables. $$ \int_{-2}^{0} \int_{0}^{x+2} e^{y^{2}-2 x y+x^{2}} d y d x $$

Let \(\mathbf{a}\) be a constant voctor and \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\). Verify the given identity. $$ \nabla \cdot[(\mathbf{r} \cdot \mathbf{r}) \mathbf{a}]=2(\mathbf{r} \cdot \mathbf{a}) $$

Find a vector that gives the direction in which the given function increases most rapidly at the indicated point. Find the maximum rate. $$ f(x, y)=x y e^{x-y} ;(5,5) $$

Graph the curve \(C\) that is described by \(\mathbf{r}\) and graph \(\mathbf{r}^{\prime}\) at the indicated value of \(\boldsymbol{t}\). $$ \mathbf{r}(t)=3 \cos t \mathbf{i}+3 \sin t \mathbf{j}+2 t \mathbf{k} ; t=\pi / 4 $$

In Problems 23-26, evaluate the given double integral by means of an appropriate change of variables. \(\iint_{R}(6 x+3 y) d A\), where \(R\) is the trapezoidal region in the first quadrant with vertices \((1,0),(4,0),(2,4)\), and \(\left(\frac{1}{2}, 1\right)\)

In Problems, evaluate the given iterated integral by changing to polar coordinates. $$ \int_{-3}^{3} \int_{0}^{\sqrt{9-x^{2}}} \sqrt{x^{2}+y^{2}} d y d x $$

$$ \text { Discuss the curvature near a point of inflection of } y=F(x) \text { . } $$

Find the volume of the solid bounded by the graphs of the given equations. $$ 2 x+y+z=6, x=0, y=0, z=0, \text { first octant } $$