Chapter 13: Q. 42 (page 1027)
Each of the integrals or integral expressions in Exercises 39-46 represents the volume of a solid in . Use polar coordinates to describe the solid, and evaluate the expressions.
Short Answer
The value of integral is
Chapter 13: Q. 42 (page 1027)
Each of the integrals or integral expressions in Exercises 39-46 represents the volume of a solid in . Use polar coordinates to describe the solid, and evaluate the expressions.
The value of integral is
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Get started for freeIn 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.
Evaluate the triple integrals over the specified rectangular solid region.
Let be a lamina in the xy-plane. Suppose is composed of n non-overlapping laminæ role="math" localid="1650321722341" Show that if the masses of these laminæ are and the centers of masses are then the center of mass of is where
Discuss the similarities and differences between the definition of the definite integral found in Chapter 4 and the definition of the double integral found in this section.
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