Question

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

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

Step-by-step solution

Step 1. Given information: 

definite integral and double integral.

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 $f\left(x\right)$over a closed interval (say [a, b]) as the number of subintervals goes to $\infty$.

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 $a\le x\le b,c\le y\le d)$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 $\infty$.