Question

Explain how the Fundamental Theorem of Calculus is used in evaluating the iterated integral ${\int }_a^b{\int }_c^{d}f(x,y)dydx$.

Short answer

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

Step 1. Given information: 

${\int }_a^b{\int }_c^{d}f(x,y)dydx$

Step 2. Explaining the Fundamental Theorem of Calculus use:

Given Iterated integral:${\int }_a^b{\int }_c^{d}f(x,y)dydx$

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.