I = $\underset{1}{\overset{3}{\int }}\left(3{\mathit{x}}^2+2\mathit{x}\right)$ dx
Here a = 1 , b = 3
f (x) = $3{\mathit{x}}^2$ + 2x
h = $\frac{\mathit{b}-\mathit{a}}{\mathit{n}}$ = $\frac{2}{\mathit{n}}$
Since, $\underset{\mathit{a}}{\overset{\mathit{b}}{\int }}$ f (x) dx = $\underset{\mathit{h}\to 0}{\lim }$ h [f (a) + f (a + h) + ... + f (a + (n - 1) h)]
So, $\underset{1}{\overset{3}{\int }}\left(3{\mathit{x}}^2+2\mathit{x}\right)$ = $\underset{\mathit{h}\to 0}{\lim }$ h [3 ${\left(1\right)}^2$ + 2 (1)) + (3 ${\left(1+\mathit{h}\right)}^2$ + 2 (1 + h) + 3 ${\left(1+2\mathit{h}\right)}^2$ + 2 (1 + 2h)) ... + 3 ${\left(1+\left(\mathit{n}-1\right)\mathit{h}\right)}^2$ + 2 (1 + (n - 1) h)]
= $\underset{\mathit{h}\to 0}{\lim }$ h [3 (n) + 3 $\left({\mathit{h}}^2+4{\mathit{h}}^2+..{\left(\mathit{n}-1\right)}^2{\mathit{h}}^2\right)$ + 3 (2h + 4h + ... + 2 (n - 1) h) + 2n + 2 (h + 2h + ... + (n - 1) h)]
= $\underset{\mathit{h}\to 0}{\lim }$ [5nh + $3{\mathit{h}}^3\left(1^2+2^2+...+{\left(\mathit{n}-1\right)}^2\right)$ + $6{\mathit{h}}^2$ (1 + 2 + ... + (n - 1)) + $2{\mathit{h}}^2$ (1 + 2 + (n - 1))]
= $\underset{\mathit{h}\to 0}{\lim }$
= $\underset{\mathit{h}\to 0}{\lim }$
= $\left[10+\frac{2\times 2\times 4}{2}+4\times 2\times 2\right]$
= 10 + 8 + 16 = 34
OR
⇒ $\frac{{\mathit{x}}^2}{9}+\frac{{\mathit{y}}^2}{4}$ = 1
⇒ y = $\frac{2}{3}\sqrt{9-{\mathit{x}}^2}$
Given line $\frac{\mathit{x}}{3}+\frac{\mathit{y}}{2}$ = 1
⇒ y = $\left(2-\frac{2\mathit{x}}{3}\right)$
Required Area $\left\{\left(\mathit{x},\mathit{y}\right):\frac{{\mathit{x}}^2}{9}+\frac{{\mathit{y}}^2}{4}\le 1\le \frac{\mathit{x}}{3}+\frac{\mathit{y}}{2}\right\}$ is given below

Required Area = $\underset{0}{\overset{3}{\int }}\left({\mathit{y}}_1-{\mathit{y}}_2\right)$ dx
= $\underset{0}{\overset{3}{\int }}\left[\frac{2}{3}\sqrt{9-{\mathit{x}}^2}-\left(2-\frac{2\mathit{x}}{3}\right)\right]$ dx
=
= $\left[\frac{2}{3}\left(\frac{9}{2}\mathit{s}\mathit{i}{\mathit{n}}^{-1}1\right)-6+3\right]$ - 0
= 3 × $\frac{\mathit{\pi }}{2}$ - 3 = $\frac{3}{2}$ (π - 2) sq units