Chapter 13: Q 14. (page 1014)

Explain why the double integral $\iint_{Ω} d A$ gives the area of the region $Ω$ . Illustrate your explanation with an example.

Short Answer

Expert verified

It is solved by solving type I integral.

Step by step solution

Given Information

We are given double integral $\iint_{Ω} d A$ over region $Ω$

Simplification

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

below by $y = g_{1} (x)$ and above by $y = g_{2} (x)$

Integral $\iint_{Ω} d A$ over type I is calculated by taking elementary area of width $∆ x$ with one end lying over $y = g_{1} (x)$ and other over $y = g_{2} (x)$

The integral becomes

$\iint_{Ω} d A = \int_{a}^{b g_{1} (x)} \int_{g_{2} (x)} d y d x$

$\iint_{Ω} d A = \int_{a}^{b} [y]_{g_{1} (x)}^{g_{2} (x)} d x$

$= \int_{a}^{b} [g_{2} (x) - g_{1} (x)] d x$

Hence, integral $\iint_{Ω} d A$ gives area over region $Ω$

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Explain why the double integral $\iint_{Ω} d A$ gives the area of the region $Ω$ . Illustrate your explanation with an example.

Short Answer

Step by step solution

Given Information

Simplification

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Explain why the double integral $\iint_{Ω} d A$ gives the area of the region $Ω$ . Illustrate your explanation with an example.