Question

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

Short answer

The Fundamental Theorem of Calculus is explained using different steps.

Step-by-step solution

Step 1. Given Information.

The iterated double integral is,

${\int }_c^d{\int }_a^{b}f\left(x,y\right)dxdy$.

Step 2. Explaining the steps.

The steps to follow are:

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