Area of the region (in square units) bounded by the curve $y=x^2+4$ and the line $y=5 x-2$ is

• A

  $\frac{1}{2}$
• B

  $\frac{1}{12}$
• C

  2
• Correct answer
  D

  $\frac{1}{6}$

Solution

We have, $$ y=x^2+4, y=5 x-2 $$ Solving equation, we get
$(2,8)$ and $(3,13)$ Area of shaded region $$ \begin{aligned} & \int_2^3\left((5 x-2)-\left(x^2+4\right) d x\right. \\ & =\int_2^3\left(5 x-x^2-6\right) d x=\left[\frac{5 x^2}{2}-\frac{x^3}{3}-6 x\right]_2^3 \\ & =\left[\left(\frac{45}{2}-9-18\right)-\left(10-\frac{8}{3}-12\right)\right]=\frac{1}{6} \end{aligned} $$

Hence, the correct answer is option: 'D'.