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Chapter 16: Problem 6

Write an integral for the average value of \(f(x, y, z)=x y z\) over the region bounded by the paraboloid \(z=9-x^{2}-y^{2}\) and the \(x y\) -plane (assuming the volume of the region is known).

Short Answer

Expert verified
Question: Determine the expression for the average value of the function f(x, y, z) = xyz over the region that is bounded by the paraboloid z = 9 - x^2 - y^2 and the xy-plane. Answer: The expression for the average value of the function f(x, y, z) = xyz over the given region is: Average value = (1/Volume) * (鈭�(from y=-3 to y=3) 鈭�(from x=-鈭�(9-y^2) to x=鈭�(9-y^2)) 鈭�(from z=0 to z=9-x^2-y^2) xyz dz dx dy)

Step by step solution

01

Determine the bounds for x, y, and z

Since the region is bounded by the xy-plane (z=0) and the paraboloid z=9-x^2-y^2, the bounds for z are [0, 9-x^2-y^2]. To find the bounds for x and y, we need to find where the paraboloid intersects the xy-plane. When z=0, we have x^2+y^2 = 9, which is the equation of a circle with radius 3 in the xy-plane. So, x will be bound by the circle equation: -鈭�(9 - y^2) <= x <= 鈭�(9 - y^2) Similarly, y will be bound by the perimeter of the circle: -3 <= y <= 3 Now that we have the bounds for x, y, and z, we can set up the triple integral.
02

Set up the triple integral

Using the bounds we found in Step 1, we will set up the triple integral as follows: Triple Integral of f(x, y, z) dV = 鈭�(from y=-3 to y=3) 鈭�(from x=-鈭�(9-y^2) to x=鈭�(9-y^2)) 鈭�(from z=0 to z=9-x^2-y^2) xyz dz dx dy
03

Evaluate the integral with respect to z

First, we will evaluate the inner integral with respect to z: 鈭�(from z=0 to z=9-x^2-y^2) xyz dz = [1/2 * xyz^2] (from z=0 to z=9-x^2-y^2) Now we are left with a double integral: 鈭�(from y=-3 to y=3) 鈭�(from x=-鈭�(9-y^2) to x=鈭�(9-y^2)) (1/2 * xy(9-x^2-y^2)^2) dx dy
04

Evaluate the double integral

The remaining double integral can be quite challenging to compute by hand. However, with the help of a symbolic or numerical computation tool (such as Wolfram Alpha, Mathcad, or Matlab), we can find the value. We won't compute the final value of the integral here. The integral for the average value of f(x, y, z) = xyz over the given region is as follows: Average value = (1/Volume) * (Triple Integral of f(x, y, z) dV) = (1/Volume) * (鈭�(from y=-3 to y=3) 鈭�(from x=-鈭�(9-y^2) to x=鈭�(9-y^2)) 鈭�(from z=0 to z=9-x^2-y^2) xyz dz dx dy)

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Most popular questions from this chapter

Sketch the following regions \(R\). Then express \(\iint_{R} g(r, \theta) d A\) as an iterated integral over \(R\) in polar coordinates. The region bounded by the spiral \(r=2 \theta,\) for \(0 \leq \theta \leq \pi,\) and the \(x\) -axis

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