The area of the region is [2025]

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Solution

Q.
The area of the region ${(x, y) : 0 \leq y \leq 2 | x | + 1, 0 \leq y \leq x^{2} + 1, | x | \leq 3}$ is [2025]

1 $\frac{17}{3}$

2 $\frac{32}{3}$

3 $\frac{64}{3}$

4 $\frac{80}{3}$

Ans.
(3)

Given, the area of region

${(x, y) : 0 \leq y \leq 2 | x | + 1, 0 \leq y \leq x^{2} + 1, | x | \leq 3}$

$∴$ Required area $= 2 \int_{0}^{2} (x^{2} + 1) d x + \int_{2}^{3} (2 x + 1) d x$

$= 2 [{(\frac{x^{3}}{3} + x)}_{0}^{2} + {(\frac{2 x^{2}}{2} + x)}_{2}^{3}]$

$= 2 [\frac{8}{3} + 2] + 2 (6)$

$= 2 [\frac{14}{3}] + 12$

$= \frac{64}{3}$ sq. units.

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