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.
Ans: Riemann sum is the basic requirement for evaluating the integral.
definite integral and double integral.
For a definite integral, equal to the limit of Riemann sum of a function of a single variable over a closed interval (say [a, b]) as the number of subintervals goes to .
Whereas, for a double integral, equal to the limit of a double Riemann sum of a function of two variables over a rectangular region defined in two directions x and y (say 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 .
Find the masses of the solids described in Exercises 53–56.
The solid bounded above by the hyperboloid with equation and bounded below by the square with vertices (2, 2, −4), (2, −2, −4), (−2, −2, −4), and (−2, 2, −4) if the density at each point is proportional to the distance of the point from the plane with equation z = −4.
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