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mathematics
calculus 4th

Questions and Answers of Calculus 4th

Calculate \(\iint_{\mathcal{S}} f(x, y, z) d S\) for the given surface and function.Part of the unit sphere centered at the origin, where \(x \geq 0\) and \(|y| \leq x ; \quad f(x, y, z)=x\)
A surface \(\mathcal{S}\) has a parametrization \(\Phi(u, v)\) with rectangular domain \(0 \leq u \leq 2,0 \leq v \leq 4\) such that the following partial derivatives are constant:\[\frac{\partial
Let \(S\) be the sphere of radius \(R\) centered at the origin. Explain using symmetry:\[\iint_{S} x^{2} d S=\iint_{S} y^{2} d S=\iint_{S} z^{2} d S\]Then show that \(\iint_{S} x^{2} d S=\frac{4}{3}
Calculate \(\iint_{\mathcal{S}}\left(x y+e^{z}ight) d S\), where \(\mathcal{S}\) is the triangle in Figure 18 with vertices \((0,0,3),(1,0,2)\), and \((0,4,1)\). (1, 0, 2) X 1 3 .(0, 0,3) 4 (0, 4, 1)
Use spherical coordinates to compute the surface area of a sphere of radius \(R\).
Use cylindrical coordinates to compute the surface area of a sphere of radius \(R\).
EAS Let \(\mathcal{S}\) be the surface with parametrization\[\Phi(u, v)=((3+\sin v) \cos u,(3+\sin v) \sin u, v)\]for \(0 \leq u \leq 2 \pi, 0 \leq v \leq 2 \pi\). Using a computer algebra system:(a)
CBS Let \(\mathcal{S}\) be the surface \(z=\ln \left(5-x^{2}-y^{2}ight)\) for \(0 \leq x \leq 1,0 \leq y \leq 1\). Using a computer algebra system:(a) calculate the surface area of \(\mathcal{S}\) to
Find the area of the portion of the plane \(2 x+3 y+4 z=28\) lying above the rectangle \(1 \leq x \leq 3,2 \leq y \leq 5\) in the \(x y\)-plane.
Use a surface integral to compute the area of that part of the plane \(a x+b y+c z=d\) corresponding to \(0 \leq x, y \leq 1\).
Find the surface area of the part of the cone \(x^{2}+y^{2}=z^{2}\) between the planes \(z=2\) and \(z=5\).
Find the surface area of the portion \(S\) of the cone \(z^{2}=x^{2}+y^{2}\), where \(z \geq 0\), contained within the cylinder \(y^{2}+z^{2} \leq 1\).
Calculate the integral of \(z e^{2 x+y}\) over the surface of the box in Figure 19. X 4 0
Calculate \(\iint_{\mathcal{S}} x^{2} z d S\), where \(\mathcal{S}\) is the cylinder (including the top and bottom) \(x^{2}+y^{2}=4,0 \leq z \leq 3\).
Let \(\mathcal{S}\) be the portion of the sphere \(x^{2}+y^{2}+z^{2}=9\), where \(1 \leq x^{2}+y^{2} \leq 4\) and \(z \geq 0\) (Figure 20). Find a parametrization of \(S\) in polar coordinates and
Prove a famous result of Archimedes: The surface area of the portion of the sphere of radius \(R\) between two horizontal planes \(z=a\) and \(z=b\) is equal to the surface area of the corresponding
Surfaces of Revolution Let \(\mathcal{S}\) be the surface formed by rotating the region under the \(\operatorname{graph} z=g(y)\) in the \(y z\)-plane for \(c \leq y \leq d\) about the \(z\)-axis,
Use Eq. (14) to compute the surface area of \(z=4-y^{2}\) for \(0 \leq y \leq 2\) rotated about the \(z\)-axis. [T+ 1+g'(y) dy area(S) = 2n
Describe the upper half of the cone \(x^{2}+y^{2}=z^{2}\) for \(0 \leq z \leq d\) as a surface of revolution (Figure 6) and use Eq. (14) to compute its surface area. area(S) = 2n /1+g'(y) dy
Area of a Torus Let \(\mathcal{T}\) be the torus obtained by rotating the circle in the \(y z\)-plane of radius \(a\) centered at \((0, b, 0)\) about the \(z\)-axis (Figure 23). We assume that
Pappus's Theorem (also called Guldin's Rule), which we introduced in Section 8.4 , states that the area of a surface of revolution \(\mathcal{S}\) is equal to the length \(L\) of the generating curve
Compute the surface area of the torus in Exercise 45 using Pappus's Theorem.Data From Exercise 45Area of a Torus Let \(\mathcal{T}\) be the torus obtained by rotating the circle in the \(y z\)-plane
Potential Due to a Uniform Sphere Let \(\mathcal{S}\) be a hollow sphere of radius \(R\) with its center at the origin with a uniform mass distribution of total mass \(m\) [since \(\mathcal{S}\) has
Calculate the gravitational potential \(V\) for a hemisphere of radius \(R\) with uniform mass distribution.
The surface of a cylinder of radius \(R\) and length \(L\) has a uniform mass distribution \(\delta\) (the top and bottom of the cylinder are excluded). Use Eq. (11) to find the gravitational
Let \(S\) be the part of the graph \(z=g(x, y)\) lying over a domain \(\mathcal{D}\) in the \(x y\)-plane. Let \(\phi=\phi(x, y)\) be the angle between the normal to \(S\) and the vertical. Prove the
Let \(\mathbf{F}\) be a vector field and \(\Phi(u, v)\) a parametrization of a surface \(\mathcal{S}\), and set \(\mathbf{N}=\mathbf{T}_{u} \times \mathbf{T}_{v}\). Which of the following is the
The vector surface integral \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\) is equal to the scalar surface integral of the function (choose the correct answer):(a) \(\|\mathbf{F}\|\).(b)
\(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\) is zero if (choose the correct answer):(a) \(\mathbf{F}\) is tangent to \(\mathcal{S}\) at every point.(b) \(\mathbf{F}\) is perpendicular to
If \(\mathbf{F}(P)=\mathbf{n}(P)\) at each point on \(\mathcal{S}\), then \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\) is equal to which of the following?(a) Zero(b)
Let \(\mathcal{S}\) be the disk \(x^{2}+y^{2} \leq 1\) in the \(x y\)-plane oriented with normal in the positive \(z\)-direction. Determine \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\) for
Estimate \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\), where \(\mathcal{S}\) is a tiny oriented surface of area 0.05 and the value of \(\mathbf{F}\) at a sample point
A small surface \(\mathcal{S}\) is divided into three pieces of area 0.2. Estimate \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\) if \(\mathbf{F}\) is a unit vector field making angles of
Let \(\mathbf{F}=\langle z, 0, yangle\), and let \(\mathcal{S}\) be the oriented surface parametrized by \(\Phi(u, v)=\left(u^{2}-v, u, v^{2}ight)\) for \(0 \leq u \leq 2\), \(-1 \leq v \leq 4\).
Let \(\mathbf{F}=\left\langle y,-x, x^{2}+y^{2}ightangle\), and let \(\mathcal{S}\) be the portion of the paraboloid \(z=x^{2}+y^{2}\) where \(x^{2}+y^{2} \leq 3\).(a) Show that if \(\mathcal{S}\) is
Suppose that \(\mathcal{S}\) is a surface in \(\mathbf{R}^{3}\) with a parametrization \(\Phi\) whose domain \(\mathcal{D}\) is the square in Figure 14. The values of a function \(f\), a vector field
Compute \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\) for the given oriented surface.\(\mathbf{F}=\langle y, z, xangle\), plane \(3 x-4 y+z=1\), \(0 \leq x \leq 1,0 \leq y \leq 1\),
Compute \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\) for the given oriented surface.\(\mathbf{F}=\left\langle e^{z}, z, xightangle, \quad \Phi(r, s)=(r s, r+s, r)\), \(0 \leq r \leq 1,0 \leq
Compute \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\) for the given oriented surface.\(\mathbf{F}=\langle 0,3, xangle\), part of sphere \(x^{2}+y^{2}+z^{2}=9\), where \(x \geq 0, y \geq 0, z
Compute \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\) for the given oriented surface.\(\mathbf{F}=\langle x, y, zangle, \quad\) part of sphere \(x^{2}+y^{2}+z^{2}=1\), where \(\frac{1}{2}
Compute \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\) for the given oriented surface.\(\mathbf{F}=\langle z, z, xangle, \quad z=9-x^{2}-y^{2}, x \geq 0, y \geq 0, z \geq 0, \quad\)
Compute \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\) for the given oriented surface.\(\mathbf{F}=\langle\sin y, \sin z, y zangle\), rectangle \(0 \leq y \leq 2,0 \leq z \leq 3\) in the \((y,
Compute \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\) for the given oriented surface.\(\mathbf{F}=y^{2} \mathbf{i}+2 \mathbf{j}-x \mathbf{k}\), portion of the plane \(x+y+z=1\) in the octant
Compute \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\) for the given oriented surface.\(\mathbf{F}=\left\langle x, y, e^{z}ightangle\), cylinder \(x^{2}+y^{2}=4,1 \leq z \leq 5\),
Compute \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\) for the given oriented surface.\(\mathbf{F}=\left\langle x z, y z, z^{-1}ightangle, \quad\) disk of radius 3 at height 4 parallel to the
Compute \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\) for the given oriented surface.\(\mathbf{F}=\langle x y, y, 0angle, \quad\) cone \(z^{2}=x^{2}+y^{2}, x^{2}+y^{2} \leq 4, z \geq 0,
Compute \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\) for the given oriented surface.\(\mathbf{F}=\left\langle 0,0, e^{y+z}ightangle, \quad\) boundary of unit cube \(0 \leq x \leq 1,0 \leq y
Compute \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\) for the given oriented surface.\(\mathbf{F}=\left\langle 0,0, z^{2}ightangle, \quad \Phi(u, v)=(u \cos v, u \sin v, v), 0 \leq u \leq 1,0
Compute \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\) for the given oriented surface.\(\mathbf{F}=\langle y, z, 0angle, \quad \Phi(u, v)=\left(u^{3}-v, u+v, v^{2}ight), 0 \leq u \leq 2,0 \leq
Let \(\mathcal{S}\) be the oriented half-cylinder in Figure 15. In (a)-(f), determine whether \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\) is positive, negative, or zero. Explain your
Let \(\mathbf{e}_{\mathbf{r}}=\langle x / r, y / r, z / rangle\) be the unit radial vector, where \(r=\sqrt{x^{2}+y^{2}+z^{2}}\). Calculate the integral of \(\mathbf{F}=\) \(e^{-r}
Show that the flux of \(\mathbf{F}=\frac{\mathbf{e}_{r}}{r^{2}}\) through a sphere centered at the origin does not depend on the radius of the sphere.
The electric field due to a point charge located at the origin in \(\mathbf{R}^{3}\) is \(\mathbf{E}=k \frac{\mathbf{e}_{r}}{r^{2}}\), where \(r=\sqrt{x^{2}+y^{2}+z^{2}}\) and \(k\) is a constant.
Let \(\mathcal{S}\) be the ellipsoid \(\left(\frac{x}{4}ight)^{2}+\left(\frac{y}{3}ight)^{2}+\left(\frac{z}{2}ight)^{2}=1\). Calculate the flux of \(\mathbf{F}=z \mathbf{i}\) over the portion of
Let \(\mathbf{v}=z \mathbf{k}\) be the velocity field (in meters per second) of a fluid in \(\mathbf{R}^{3}\). Calculate the flow rate (in cubic meters per second) through the upper hemisphere \((z
Calculate the flow rate of a fluid with velocity field \(\mathbf{v}=\left\langle x, y, x^{2} yightangle\) (in meters per second) through the portion of the ellipse
A net is dipped in a river. Determine the flow rate of water across the net if the velocity vector field for the river is given by \(\mathbf{v}\) and the net is described by the given
A net is dipped in a river. Determine the flow rate of water across the net if the velocity vector field for the river is given by \(\mathbf{v}\) and the net is described by the given
A net is dipped in a river. Determine the flow rate of water across the net if the velocity vector field for the river is given by \(\mathbf{v}\) and the net is described by the given
A net is dipped in a river. Determine the flow rate of water across the net if the velocity vector field for the river is given by \(\mathbf{v}\) and the net is described by the given
Let \(\mathcal{T}\) be the triangular region with vertices \((1,0,0),(0,1,0)\), and \((0,0,1)\) oriented with upward-pointing normal vector (Figure 16). Assume distances are in meters.A fluid flows
Let \(\mathcal{T}\) be the triangular region with vertices \((1,0,0),(0,1,0)\), and \((0,0,1)\) oriented with upward-pointing normal vector (Figure 16). Assume distances are in meters.Calculate the
Prove that if \(\mathcal{S}\) is the part of a graph \(z=g(x, y)\) lying over a domain \(\mathcal{D}\) in the \(x y\)-plane, then\[\iint_{\mathcal{S}} \mathbf{F} \cdot d
A varying current i(t) flows through a long, straight wire in the xy-plane as in Example 5. The current produces a magnetic field \(\mathbf{B}\) whose magnitude at a distance \(r\) from the wire is
Assume that \(i=10 e^{-0.1 t} \mathrm{~A}(t\) in seconds). Calculate the flux \(\Phi(t)\), at time \(t\), of \(\mathbf{B}\) through the isosceles triangle of base \(12 \mathrm{~cm}\) and height \(6
A solid material that has thermal conductivity \(K\) in kilowatts per meter-kelvin and temperature given at each point by \(w(x, y, z)\) has heat flow given by the vector field \(\mathbf{F}=-K abla
A solid material that has thermal conductivity KK in kilowatts per meter-kelvin and temperature given at each point by w(x,y,z)w(x,y,z) has heat flow given by the vector field F=−K∇wF=−K∇w
A point mass \(m\) is located at the origin. Let \(Q\) be the flux of the gravitational field \(\mathbf{F}=-G m \frac{\mathbf{e}_{r}}{r^{2}}\) through the cylinder \(x^{2}+y^{2}=R^{2}\) for \(a \leq
Let \(\mathcal{S}\) be the surface with parametrization\[\Phi(u, v)=\left(\left(1+v \cos \frac{u}{2}ight) \cos u,\left(1+v \cos \frac{u}{2}ight) \sin u, v \sin \frac{u}{2}ight)\]for \(0 \leq u \leq 2
- \(A 5\) We cannot integrate vector fields over \(\mathcal{S}\) because \(\mathcal{S}\) is not orientable, but it is possible to integrate functions over \(\mathcal{S}\). Using a computer algebra
Let D be a disk in R2. This exercise shows that if Vf(x, y) = 0 for all (x, y) in D, then f is constant. Consider points P = (a,b), Q = (c,d), and R = (c, b) as in Figure 16. (a) Use single-variable
Let ϕ = ln r, where r = √x2 + y2. Express ∇ϕ in terms of the unit radial vector er in R2.
For P = (a, b), we define the unit radial vector field based at P:(a) Verify that eP is a unit vector field.(b) Calculate eP(1, 1) for P = (3, 2).(c) Find a potential function for eP. ep = (x-a,y -
Which of (A) or (B) in Figure 12 is the contour plot of a potential function for the vector field F? Recall that the gradient vectors are perpendicular to the level curves. y (A) X y > X y (B) x
Which of (A) or (B) in Figure 13 is the contour plot of a potential function for the vector field F?
Match each of these descriptions with a vector field in Figure 14.(a) The gravitational field created by two planets of equal mass located at P and Q(b) The electrostatic field created by two equal
In this exercise, we show that the vector field F in Figure 15 is not conservative. Explain the following statements:(a) If a potential function ƒ for F exists, then the level curves of ƒ must be
Show that any vector field of the form F = (ƒ(x), g(y), h(z)) has a potential function. Assume that ƒ, g, and h are continuous.
What is the line integral of the constant function ƒ(x, y, z) = 10 over a curve C of length 5?
Which of the following have a zero line integral over the vertical segment from (0, 0) to (0, 1)? (a) f(x, y) = x (c) F = (x, 0) (e) F = (0, x) (b) f(x, y) = y (d) F = (y,0) (f) F = (0, y)
State whether each statement is true or false. If the statement is false, give the correct statement.(a) The scalar line integral does not depend on how you parametrize the curve.(b) If you reverse
Suppose that C has length 5. What is the value of (a) F(P) is normal to C at all points P on C?(b) F(P) is a unit vector pointing in the negative direction along the curve? So F. dr if:
Let \(f(x, y, z)=x+y z\), and let \(C\) be the line segment from \(P=(0,0,0)\) to \((6,2,2)\).(a) Calculate \(f(\mathbf{r}(t))\) and \(d s=\left\|\mathbf{r}^{\prime}(t)ight\| d t\) for the
Repeat Exercise 1 with the parametrization \(\mathbf{r}(t)=\left\langle 3 t^{2}, t^{2}, t^{2}ightangle\) for \(0 \leq t \leq \sqrt{2}\).Data From Exercise 1Let \(f(x, y, z)=x+y z\), and let \(C\) be
Let \(\mathbf{F}=\left\langle y^{2}, x^{2}ightangle\), and let \(C\) be the curve \(y=x^{-1}\) for \(1 \leq x \leq 2\), oriented from left to right.(a) Calculate \(\mathbf{F}(\mathbf{r}(t))\) and \(d
Let \(\mathbf{F}(x, y, z)=\left\langle z^{2}, x, yightangle\), and let \(C\) be the curve that is given by \(\mathbf{r}(t)=\left\langle 3+5 t^{2}, 3-t^{2}, tightangle\) for \(0 \leq t \leq 2\).(a)
Compute the integral of the scalar function or vector field over \(\mathbf{r}(t)=\langle\cos t, \sin t, tangle\) for \(0 \leq t \leq \pi\).\(f(x, y, z)=x^{2}+y^{2}+z^{2}\)
Compute the integral of the scalar function or vector field over \(\mathbf{r}(t)=\langle\cos t, \sin t, tangle\) for \(0 \leq t \leq \pi\).\(f(x, y, z)=x y+z\)
Compute the integral of the scalar function or vector field over \(\mathbf{r}(t)=\langle\cos t, \sin t, tangle\) for \(0 \leq t \leq \pi\).\(\mathbf{F}(x, y, z)=\left\langle x, y, z^{2}ightangle\)
Compute the integral of the scalar function or vector field over \(\mathbf{r}(t)=\langle\cos t, \sin t, tangle\) for \(0 \leq t \leq \pi\).\(\mathbf{F}(x, y, z)=\left\langle x y, 2, z^{3}ightangle\)
Compute \(\int_{C} f d s\) for the curve specified.\(f(x, y)=\sqrt{1+9 x y}, \quad y=x^{3}\) for \(0 \leq x \leq 2\)
Compute \(\int_{C} f d s\) for the curve specified.\(f(x, y)=\frac{y^{3}}{x^{7}}, \quad y=\frac{1}{4} x^{4}\) for \(1 \leq x \leq 2\)
Compute \(\int_{C} f d s\) for the curve specified.\(f(x, y, z)=z^{2}, \quad \mathbf{r}(t)=\langle 2 t, 3 t, 4 tangle\) for \(0 \leq t \leq 2\)
Compute \(\int_{C} f d s\) for the curve specified.\(f(x, y, z)=3 x-2 y+z, \quad \mathbf{r}(t)=\langle 2+t, 2-t, 2 tangle\) for \(-2 \leq t \leq 1\)
Compute \(\int_{C} f d s\) for the curve specified.\(f(x, y, z)=x e^{z^{2}}\), piecewise linear path from \((0,0,1)\) to \((0,2,0)\) to \((1,1,1)\)
Compute \(\int_{C} f d s\) for the curve specified.\(f(x, y, z)=x^{2} z, \quad \mathbf{r}(t)=\left\langle e^{t}, \sqrt{2} t, e^{-t}ightangle\) for \(0 \leq t \leq 1\)
Compute \(\int_{C} f d s\) for the curve specified.\(f(x, y, z)=2 x^{2}+8 z, \quad \mathbf{r}(t)=\left\langle e^{t}, t^{2}, tightangle, \quad 0 \leq t \leq 1\)
Compute \(\int_{C} f d s\) for the curve specified.\(f(x, y, z)=6 x z-2 y^{2}, \quad \mathbf{r}(t)=\left\langle t, \frac{t^{2}}{\sqrt{2}}, \frac{t^{3}}{3}ightangle, \quad 0 \leq t \leq 2\)
Calculate \(\int_{C} 1 d s\), where the curve \(C\) is parametrized by \(\mathbf{r}(t)=\langle 4 t,-3 t, 12 tangle\) for \(2 \leq t \leq 5\). What does this integral represent?
Calculate \(\int_{C} 1 d s\), where the curve \(C\) is parametrized by \(\mathbf{r}(t)=\left\langle e^{t}, \sqrt{2} t, e^{-t}ightangle\) for \(0 \leq t \leq 2\).

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