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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 $[0, \frac{1}{2}] 脳 [0, \frac{1}{3}]$ in the XY-plane.

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Step by step solution

Step 1. Given information.

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

Step 2.

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Most popular questions from this chapter

What are the outputs of a vector field in the Cartesian plane?

$F (x, y, z) = 鉄� y z, x z, x y 鉄�$ , where S is the portion of the saddle determined by $z = x^{2} 鈭� y^{2}$ that lies above the region in thexy-plane bounded by the x-axis and the parabola with equation $y = 1 鈭� x^{2}$ .

Integrate the given function over the accompanying surface in Exercises 27鈥�34.

$f (x, z) = e^{- (x^{2} + z^{2})}$ , where S is the unit disk centered at the point (0, 2, 0)and in the plane y = 2.

Compute the curl of the vector fields:

$F (x, y) = \cos (x + y) i + \sin (x - y) j$ .

Give an example of a vector field whose orientation does not affect the outcome of Stokes鈥� Theorem.

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Find the integral of f(x,y,z)=z3+zx2+2yon the portion of the unit sphere that lies in the first octant, above the rectangle 0,12脳0,13 in the XY-plane.

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Step 1. Given information.

Step 2.

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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 $[0, \frac{1}{2}] 脳 [0, \frac{1}{3}]$ in the XY-plane.