Question

Let $g_1\left(x\right)\text{and}{g}_2\left(x\right)$be two continuous functions such that $g_1\left(x\right)\le g_2\left(x\right)$on the interval $[a,b]$, and let be the region in the $xy$-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${\int }_a^{b{g}_{g_1}\left(x\right)}{\int }_2^{g_2\left(x\right)}dydx$

Step-by-step solution

Given Information

Two continuous functions $g_1\left(x\right)\text{and}{g}_2\left(x\right)$such that $g_1\left(x\right)<g_2\left(x\right)$in $\left[a,b\right]$.

Region $\Omega$is bounded by curves $g_1\text{and}{g}_2$in $\left[a,b\right]$.

Consideration

The region bounded on left by $x=a$and right by $x=a$, $y=g_1\left(x\right)$below and $y=g_1\left(x\right)$above is of type I

Hence, surface integral becomes

${\iint }_{\Omega }dA={\int }_a^{b{g}_{g_1}\left(x\right)}{\int }_2^{g_2\left(x\right)}dydx$

Integral that gives area of region is ${\int }_a^b{\int }_{g_1\left(x\right)}^{g_2\left(x\right)}dydx$.

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

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

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