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Let \(\mathbf{e}_r = \left\langle \frac{x}{r}, \frac{y}{r}, \frac{z}{r} \right\rangle\) be a unit radial vector, where \(r = \sqrt{x^2 + y^2 + z^2}\). (a) Calculate the integral of \(\mathbf{F} = e^{-r}\mathbf{e}_r\) over the upper hemisphere of \(x^2 + y^2 + z^2 = 9\) with the normal pointing outward. (Give your answer in exact form. Use symbolic notation and fractions where needed.) \(\iint_S \mathbf{F} \cdot d\mathbf{S} = 18\pi e^{-3}\) (b) Calculate the integral of \(\mathbf{F} = 3e^{-r}\mathbf{e}_r\) over the octant \(x \ge 0, y \ge 0, z \ge 0\) of the unit sphere centered at the origin. (Give your answer in exact form. Use symbolic notation and fractions where needed.) \(\iint_S \mathbf{F} \cdot d\mathbf{S} = 54\pi e^{-3}\)

          Let \(\mathbf{e}_r = \left\langle \frac{x}{r}, \frac{y}{r}, \frac{z}{r} \right\rangle\) be a unit radial vector, where \(r = \sqrt{x^2 + y^2 + z^2}\).
(a) Calculate the integral of \(\mathbf{F} = e^{-r}\mathbf{e}_r\) over the upper hemisphere of \(x^2 + y^2 + z^2 = 9\) with the normal pointing outward.
(Give your answer in exact form. Use symbolic notation and fractions where needed.)
\(\iint_S \mathbf{F} \cdot d\mathbf{S} = 18\pi e^{-3}\)
(b) Calculate the integral of \(\mathbf{F} = 3e^{-r}\mathbf{e}_r\) over the octant \(x \ge 0, y \ge 0, z \ge 0\) of the unit sphere centered at the origin.
(Give your answer in exact form. Use symbolic notation and fractions where needed.)
\(\iint_S \mathbf{F} \cdot d\mathbf{S} = 54\pi e^{-3}\)

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$Let 𝐞r = ⟨(x)/(r), (y)/(r), (z)/(r)⟩ be a unit radial vector, where r = √(x^2 + y^2 + z^2). (a) Calculate the integral of 𝐅 = e^-r𝐞r over the upper hemisphere of x^2 + y^2 + z^2 = 9 with the normal pointing outward. (Give your answer in exact form. Use symbolic notation and fractions where needed.) 𝐅· d𝐒 = 18π e^-3 (b) Calculate the integral of 𝐅 = 3e^-r𝐞r over the octant x ≥ 0, y ≥ 0, z ≥ 0 of the unit sphere centered at the origin. (Give your answer in exact form. Use symbolic notation and fractions where needed.) 𝐅· d𝐒 = 54π e^-3$

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Calculus: Early Transcendentals

James Stewart 8th Edition

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Solved by Expert Linda Winkler

12/11/2021

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In spherical coordinates, the unit radial vector $\mathbf{r}$ can be written as $\mathbf{r}=\sin\theta\cos\phi\mathbf{i}+\sin\theta\sin\phi\mathbf{j}+\cos\theta\mathbf{k}$, where $\theta$ is the polar angle and $\phi$ is the azimuthal angle. The equation of the Show more…

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Let $\mathbf{r}$ be a unit radial vector, where $r=x^2+y^2+z^2$. a) Calculate the integral of $\mathbf{F}=e^{-\mathbf{r}}$ over the upper hemisphere of $x^2+ y^2+ z^2=9$ with the normal pointing outward. (Give your answer in exact form. Use symbolic notation and fractions where needed.) b) Calculate the integral of $\mathbf{F}=3e^{-\mathbf{r}}$ over the octant $x>0, y>0, z>0$ of the unit sphere centered at the origin. (Give your answer in exact form. Use symbolic notation and fractions where needed.)

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00:01 So taking the surface integral of a vector field, a vector field typically has three components, and a surface has not only an area associated with it, but a direction.

00:21 And what we'd like to demonstrate here is that sometimes one of these surface intervals looks scary, but if you use the right variables, it becomes very easy.

00:35 So here we have a hemispherical surface.

00:40 It is the upper hemisphere of a sphere with a radius equal to 9.

00:49 And what we know for sure, sorry, radius is equal to 3, square root of 9.

00:54 But what we know for sure is that the radius is constant along that surface.

01:00 So what we mean by the surface, if we wanted to write it, it would involve two spherical angles.

01:10 I usually use theta as the co -latitude and phi as a mutual angle.

01:18 So let me kind of just show what i typically use.

01:22 Theta goes down from the north pole and phi goes flying around the sphere around the pole, if you will.

01:34 But it points in the radially outward direction.

01:39 We'll call that er.

01:42 And so if your vector function does not depend on those angles, basically what you will get is the surface area from the angle.

01:57 And in this case, recognize that a full sphere would have a surface area equal to 4 pi r squared.

02:10 And here we only have 2 pi r squared because it's the hemisphere.

02:24 And our vector function, because it depends just on the radius, r is constant as well as the projection of the force onto the surface area.

02:40 So basically what we have is the surface area of the sphere times the radial component of the vector field evaluated on the surface.

02:54 And that's all we have.

02:58 So putting this all together, we have 2 pi.

03:02 Our radius is 3.

03:04 So that's 3 squared.

03:06 And then e to the minus 3...

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