Chapter 13: Q.25 (page 1082)
Evaluating triple integrals: Each of the triple integrals that follow represents the volume of a solid. Sketch the solid and evaluate the integral.
Short Answer
The value of the triple integral is 30.
Chapter 13: Q.25 (page 1082)
Evaluating triple integrals: Each of the triple integrals that follow represents the volume of a solid. Sketch the solid and evaluate the integral.
The value of the triple integral is 30.
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Get started for freeIdentify the quantities determined by the integral expressions in Exercises 19–24. If x, y, and z are all measured in centimeters and ρ(x, y,z) is a density function in grams per cubic centimeter on the three-dimensional region , give the units of the expression.
Find the signed volume between the graph of the given function and the xy-plane over the specified rectangle in the xy-plane
Identify the quantities determined by the integral expressions in Exercises 19–24. If x, y, and z are all measured in centimeters and ρ(x, y,z) is a density function in grams per cubic centimeter on the three-dimensional region , give the units of the expression.
Let be an integrable function on the rectangular solid , and let Use the definition of the triple integral to prove that:
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.
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