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Chapter 13: Q. 9 (page 1003)

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.

Short Answer

Expert verified

Ans: Riemann sum is the basic requirement for evaluating the integral.

Step by step solution

01

Step 1. Given information: 

definite integral and double integral.

02

Step 2. Discussing the similarities and differences between the definition of the definite integral and double integral:

For a definite integral, equal to the limit of Riemann sum of a function of a single variable f(x)over a closed interval (say [a, b]) as the number of subintervals goes to 鈭�.

Whereas, for a double integral, equal to the limit of a double Riemann sum of a function of two variables f(x,y)over a rectangular region defined in two directions x and y (say a鈮�x鈮�b,c鈮�y鈮�d)that has been subdivided into smaller subrectangles in both the x and y directions as the numbers of subintervals in those two directions goes to 鈭�.

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Most popular questions from this chapter

Find the volume between the graph of the given function and the xy-plane over the specified rectangle in the xy-plane

f(x,y)=-2x2y3WhereR={(x,y)|-2鈮�x鈮�3,-1鈮�y鈮�5}

Find the masses of the solids described in Exercises 53鈥�56.

The solid bounded above by the plane with equation 2x + 3y 鈭� z = 2 and bounded below by the triangle with vertices (1, 0, 0), (4, 0, 0), and (0, 2, 0) if the density at each point is proportional to the distance of the point from the

xy-plane.

In the following lamina, all angles are right angles and the density is constant:

LetR=x,y,z|a1猢�x猢�a2,b1猢�y猢�b2,c1猢�z猢�c2.Provethat:鈭�RdV=a2-a1b2-b1c2-c1.WhatistherelationshipbetweenRandtheproducta2-a1b2-b1c2-c1.

In Exercises 57鈥�60, let R be the rectangular solid defined by

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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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