The area (in sq. units) of the region

Question:

The area (in sq. units) of the region

$\left\{(x, y): 0 \leq y \leq x^{2}+1,0 \leq y \leq x+1\right.$

$\left.\frac{1}{2} \leq x \leq 2\right\}$ is :

  1. $\frac{79}{16}$

  2. $\frac{23}{6}$

  3. $\frac{79}{24}$

  4. $\frac{23}{16}$


Correct Option: , 3

Solution:

$0 \leq y \leq x^{2}+1,0 \leq y \leq x+1, \frac{1}{2} \leq x \leq 2$

Required area $=\int_{1 / 2}^{1}\left(x^{2}+1\right) d x+\frac{1}{2}(2+3) \times 1$

$=\frac{19}{24}+\frac{5}{2}=\frac{79}{24}$

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