Chapter 13: Q. 25 (page 1024)
Each of the integrals or integral expressions in Exercise represents the area of a region in the plane. Use polar coordinates to sketch the region and evaluate the expressions.
Short Answer
The integral's value is
Chapter 13: Q. 25 (page 1024)
Each of the integrals or integral expressions in Exercise represents the area of a region in the plane. Use polar coordinates to sketch the region and evaluate the expressions.
The integral's value is
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Get started for freeDescribe the three-dimensional region expressed in each iterated integral in Exercises 35–44.
Find the masses of the solids described in Exercises 53–56.
The first-octant solid bounded by the coordinate planes and the plane 3x + 4y + 6z = 12 if the density at each point is proportional to the distance of the point from the xz-plane.
In Exercises 57–60, let R be the rectangular solid defined by
R = {(x, y, z) | 0 ≤ x ≤ 4, 0 ≤ y ≤ 3, 0 ≤ z ≤ 2}.
Assume that the density ofR is uniform throughout.
(a) Without using calculus, explain why the center of mass is (2, 3/2, 1).
(b) Verify that the center of mass is (2, 3/2, 1), using the appropriate integral expressions.
Earlier in this section, we showed that we could use Fubini’s theorem to evaluate the integral and we showed that Now evaluate the double integral by evaluating the iterated integral.
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