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  • please solve RD sharma class 12 chapter 20 Area of bounded Region exercise 20.4 question 2 maths textbook solution

please solve RD sharma class 12 chapter 20 Area of bounded Region exercise 20.4 question 2 maths textbook solution

Answers (1)

\frac{92}{3} sq.units

Hint:

Find the point of intersection of parabola and the given line and then integrate it to find the required region.

Given:

               x=8+2y-y^{2},y=-1 and y=3

Solution:

The parabola cuts y-axis at (0,4) and (0,-2)

Also, the points of intersection of the parabola and the line y=-1 and y=3 are B(5,3) and D(5,-1) respectively.

The area required is the region ABCDE

\begin{aligned} A &=\int_{-1}^{3} x d y \\ &=\int_{-1}^{3}\left(8+2 y-y^{2}\right) d y \\ &=\left[8 y+y^{2}-\frac{y^{3}}{3}\right]_{-1}^{3} \\ \end{aligned}

\begin{aligned} &=\left[8(3)+3^{2}-\frac{3^{3}}{3}\right]-\left[8(-1)+(-1)^{2}-\frac{(-1)^{3}}{3}\right] \\ \end{aligned}

\begin{aligned} &=24+9-9+8-1-\frac{1}{3} \\ &=31-\frac{1}{3} \\ &=\frac{92}{3} s q \cdot \text { units } \end{aligned}

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