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Chapter 13: Q. 15 (page 1003)

Explain how the Fundamental Theorem of Calculus is used in evaluating the iterated integral 鈭�ab鈭�cdf(x,y)dydx.

Short Answer

Expert verified

Ans:

part (a). Find an anti-derivative.

part (b). Use the fundamental theorem of calculus.

part (c). Now the resultant will be the definite integral.

part (d). Find the anti-derivative of the function.

part (e). Evaluating the function and evaluating the difference between them gives the final result.

Step by step solution

01

Step 1. Given information: 

鈭�ab鈭�cdf(x,y)dydx

02

Step 2. Explaining the Fundamental Theorem of Calculus use:

Given Iterated integral:鈭�ab鈭�cdf(x,y)dydx

For Finding the given iterated double integral using the Fundamental Theorem of Calculus, the steps to follow are:

  • Find an anti-derivative of f(x,y)with respect to y.
  • Use the fundamental theorem of calculus to evaluate the inner integral by evaluating the function from step 1 at d and c and evaluating the difference between them.
  • Now the resultant will be the definite integral of a function with a single variable x.
  • Find the anti-derivative of the function of step 3 with respect to x.
  • Evaluating the function of step 4 at b and a and evaluating the difference between them gives the final result of the double integral of the given function.

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