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 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 at each point in Ris proportional to the distance of the point from the xy-plane.
(a) Without using calculus, explain why the x- and y-coordinates of the center of mass are respectively.
(b) Use an appropriate integral expression to find the z-coordinate of the center of mass.
Let be a lamina in the xy-plane. Suppose is composed of two non-overlapping lamin and , as follows:
Show that if the masses and centers of masses of and are and and respectively, then the center of mass of is where
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.
Explain how the Fundamental Theorem of Calculus is used in evaluating the iterated integral.
Evaluate each of the double integral in the exercise 37-54 as iterated integrals
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