Question

Let $h_1\left(y\right)\text{and}{h}_2\left(y\right)$be two continuous functions such that $g_1\left(x\right)\le g_2\left(x\right)\text{on}[a,b]$and let $\Omega$be region in $xy$plane bounded by $g_1\text{and}{g}_2\text{on}[a,b]$. Use your answer to Exercise 14 to set up an iterated integral whose value is the area of $\Omega$. How is this iterated integral related

to the definite integral you would have used to compute the area of $\Omega$in Chapter 4?

Short answer

The iterated integral that gives area of region is${\int }_c^{d{h}_2\left(y\right)}{\int }^{h_2}dxdy$

Step-by-step solution

Given Information

Two continuous functions $h_1\left(y\right)\text{and}{h}_2\left(y\right)$are given such that $h_1\left(y\right)<h_2\left(y\right)$, in $\left[c,d\right]$

Assume$\Omega$be region in$xy$plane bounded by curves$h_1\text{and}{h}_2$in$\left[c,d\right]$

Simplification

Consider arbitrary type II region.

The integral ${\iint }_{\Omega }dA$type type II is calculated by taking elemental area of width $\Delta y$with left end lying over $x=h_1\left(y\right)$and right end lying over $x=h_2\left(y\right)$

Surface integral becomes

${\iint }_{\Omega }dA={\int }_{c{h}_1\left(y\right)}^{d{h}_2\left(y\right)}dxdy$

${\iint }_{\Omega }dA={\int }_c^d[x{]}_{h_1\left(y\right)}^{h_2\left(y\right)}dy$

$={\int }_c^d\left[h_2\left(y\right)-h_1\left(y\right)\right]dy$

Hence, the iterated integral that gives area of region is${\int }_c^{d{h}_2\left(y\right)}{\int }_1^{h_2\left(y\right)}dxdy$