Q 20. When you wish to evaluate the de... [FREE SOLUTION]

Chapter 13: Q 20. (page 1014)

When you wish to evaluate the definite integral ${鈭�}_{a}^{b} f (x) d x$ of continuous function $f$ , interval $[a, b]$ is never an impediment to using the Fundamental Theorem of Calculus. However, when you wish to evaluate the double integral ${鈭�}_{惟} g (x, y) d A$ of a continuous function $g$ over $惟$ , the region can make the evaluation process easier or harder. Why?

Expert verified

It would be difficult to evaluate functions wrt $x$ .

We need to find difference between definite integral ${鈭�}_{a}^{b} f (x) d x$ and double integral ${鈭�}_{a}^{b} f (x) d x$ .

$f$ is continuous function over $[a, b]$ and $g$ over $惟$ .

Definite integral is calculated using Fundamental Theorem of Calculus.

For calculation of double integral, ${鈭�}_{惟} g (x, y) d A$ , integration is performed wrt $x$ or wrt $y$ according to type of region.

After first integral, it may be easy or hard.

Assume integral is evaluated wrt $x$ first.

The integral along with mentioned limits are

${鈭�}_{惟} g (x, y) d A = {鈭�}_{c}^{d h_{2} (x)} {鈭�}_{h_{1} (x)} g (x, y) d y d x$

Solving wrt $y$ first

${鈭�}_{惟} g (x, y) d A = {鈭�}_{c}^{d} [G (x)]_{h_{2} (x)}^{h_{2} (x)} d x$

$= {鈭�}_{c}^{d} {[G (h_{2} (x)) - G (h_{1} (x))]}_{h_{1} (x)}^{h_{2} (x)} d x$

and $G (x) = 鈭� g (x, y) d y$

Functions $G (h_{2} (x)) and G (h_{1} (x))$ can be difficult to evaluate wrt $x$

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