Q 14. - Calculus

Question

Explain why the double integral ${\iint }_{\Omega }dA$ gives the area of the region $\Omega$ . Illustrate your explanation with an example.

Step-by-Step Solution

Answer

It is solved by solving type I integral.

Step 1: Given Information

We are given double integral ${\iint }_{\Omega }dA$ over region $\Omega$

Step 2: Simplification

Consider arbitrary type I region bounded on left by $x=a$, to right by $x=b$

below by $y=g_1\left(x\right)$ and above by $y=g_2\left(x\right)$

Integral ${\iint }_{\Omega }dA$ over type I is calculated by taking elementary area of width $∆x$ with one end lying over $y=g_1\left(x\right)$ and other over $y=g_2\left(x\right)$

The integral becomes

${\iint }_{\Omega }dA={\int }_a^{b{g}_1\left(x\right)}{\int }_{g_2\left(x\right)}dydx$

${\iint }_{\Omega }dA={\int }_a^b[y{]}_{g_1\left(x\right)}^{g_2\left(x\right)}dx$

$={\int }_a^b\left[g_2\left(x\right)-g_1\left(x\right)\right]dx$

Hence, integral ${\iint }_{\Omega }dA$ gives area over region $\Omega$