Q 3. Explain the difference between a... [FREE SOLUTION]

Chapter 13: Q 3. (page 1014)

Explain the difference between a type I region and a type II region.

Short Answer

Expert verified

It can be seen that, in the case of type I region, the integration is first done with respect to $y$ and then with respect to $x$ .

Step by step solution

Given Information

Two regions are given. We need to find the difference between two.

Consideration

In interval $[a, b], a < b$ , type I region is bounded by $y = g_{1} (x)$ at the bottom and $y = g_{2} (x)$ at top for all $x 鈭� [a, b]$ .

In interval $[c, d], c < d$ , type I region is bounded by $x = h_{1} (y)$ to the left and $x = h_{2} (y)$ to the right for all $y 鈭� [c, d]$

Simplification of first region

The surface integral is calculated as ${鈭�}_{惟} f (x, y) d A$

It can be calculated by taking an elemental area of breadth $螖 x$ , one end lying on $y = g_{1} (x)$ and other on $y = g_{2} (x)$

Hence, surface integral is

${鈭�}_{惟} f (x, y) d A = {鈭�}_{a}^{b g_{1}} {鈭�}_{g_{1} (x)}^{g_{2} (x)} f (x, y) d y d x$

Simplification of second region

It can be calculated by taking an elemental area of breadth $螖 y$ one end lying on $x = h_{1} (y)$ and other on role="math" localid="1653920549681" $x = h_{2} (y)$

The integral becomes

${鈭�}_{惟} f (x, y) d A = {鈭�}_{c h_{4} (y)}^{d h_{2} (y)} f (x, y) d x d y$

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Explain the difference between a type I region and a type II region.

Short Answer

Step by step solution

Given Information

Consideration

Simplification of first region

Simplification of second region

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