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The area (in sq. units) of the region $A=\left \{ (x,y)\epsilon R\times R\left |0 \leq x\leq 3,0\leq y\leq 4,y\leq x^{2}+3x \right \}$ is :

Option 1)
$\frac{53}{6}$
Option 2)
$\frac{59}{6}$
Option 3)
$\frac{26}{3}$

Option 4)
$8$

Answers (1)

S solutionqc

$A=\left \{ (x,y)\epsilon R\times R\left |0 \leq x\leq 3,0\leq y\leq 4,y\leq x^{2}+3x \right \}$

$=y=A$

Area of OCD $x=1$

$\int_{0}^{1}(4-x^{2}-3x)d\; x=\; 4x-\frac{x^{3}}{3}-\frac{3x^{2}}{2}\mid _{0}^{1}$

$=4\times1-\frac{1}{3}-\frac{3}{2}-0$

$=4-\frac{1}{3}-\frac{3}{2}$

$=4-\left ( \frac{2+3}{6} \right )$

$=\frac{24-11}{6}=\frac{13}{6}$

Area of DA BD - Area of OCD $=3\times4-\frac{13}{6}$

$=\frac{72-13}{6}=\frac{59}{6}$

Option 1)

$\frac{53}{6}$

Option 2)
$\frac{59}{6}$
Option 3)

$\frac{26}{3}$

Option 4)

$8$

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