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Chapter 14: Q. 45 (page 1120)

Find the integral of $f (x, y, z) = z^{3} + z (x^{2} + 2^{y})$ on the portion of the unit sphere that lies in the first octant, above the rectangle $[0, \frac{1}{2}] \times [0, \frac{1}{3}]$ in the XY-plane.

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Step by step solution

01

Step 1. Given information.

give, $[0, \frac{1}{2}] \times [0, \frac{1}{3}]$

02

Step 2.

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Most popular questions from this chapter

Use the curl form of Green’s Theorem to write the line integral of F(x, y) about the unit circle as a double integral. Do not evaluate the integral.

Problem Zero: Read the section and make your own summary of the material.

$Compute the divergence of the vector fields in Exercises 17 - 22 . F (x, y, z) = {xe}^{yz} i + {ye}^{xz} j + {ze}^{xy} k$

The current through a certain passage of the San Juan Islands in Washington State is given by

$F = <F_{1} (x, y), F_{2} (x, y)> = <0, 1.152 - 0.8 x^{2}> .$

Consider a disk R of radius 1 mile and centered on this region. Denote the boundary of the disk by localid="1650455155947" $\partial R$ .

(a) Compute localid="1650455166518" $\iint_{R} (\frac{\partial F_{2}}{\partial x} - \frac{\partial F_{1}}{\partial y}) d A$ .
(b) Show that $\int_{\partial R} F \cdot n d s = \int_{0}^{2 π} (1.152 - 0.8 \cos^{2} θ) \sin θ d θ .$ Conclude that Green's Theorem is valid for the current in this area of the San Juan Islands.
(c) What do the integrals from Green's Theorem tell us about this region of the San Juan Islands?

Give a smooth parametrization, in terms of u and v, of the sphere of radius k and centered at the origin.

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