Chapter 13: Q 65. (page 1014)
Let be an integrable function on the rectangle and , and let . Use the definition of the double integral to prove that
Short Answer
To prove this, write the double integral on left hand side as double Reimann sum.
Chapter 13: Q 65. (page 1014)
Let be an integrable function on the rectangle and , and let . Use the definition of the double integral to prove that
To prove this, write the double integral on left hand side as double Reimann sum.
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Get started for freeExplain how the Fundamental Theorem of Calculus is used in evaluating the iterated integral .
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.
Use the results of Exercises 59 and 60 to find the centers of masses of the laminæ in Exercises 61–67.
Use the lamina from Exercise 61, but assume that the density is proportional to the distance from the x-axis.
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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 of R is uniform throughout, and find the moment of inertia about the x-axis and the radius of gyration about the x-axis.
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