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  • Need solution for RD Sharma Maths Class 12 Chapter 20 Areas of Bounded Regions exercise 20.3 question 29.

Need solution for RD Sharma Maths Class 12 Chapter 20 Areas of Bounded Regions exercise 20.3 question 29.

Answers (1)

Answer:

\frac{9}{2}\text{sq. units}

Hint:

 Use concept.

Given:

The given equations are y=2-x^2 and  y+x=0 .

Solution:

 To find area bounded by

y=2-x^2            ......(1)

y+x=0        .......(2)

Equation (1) represents  a parabola with vertex (0,2)  and downward, meets axes at (\pm 0, \sqrt{2}) .

Equation (2) represents a line passing through (0,0)   and (2,-2) .The points of intersection of line  and parabola are  (2,-2)  and (1,-1) .

A rough sketch of curve is as follows:

Shaded region is sliced into rectangles with area =(y_2-y_1)_{\Delta }x. It slides from x =-1  to x = 2 so,

Required area =Region ABPCOA

\begin{aligned} &A=\int_{-1}^{2}\left(y_{1}-y_{2}\right) d x \\ &=\int_{-1}^{2}\left(2-x^{2}+x\right) d x \\ &=\left[2 x-\frac{x^{3}}{3}+\frac{x^{2}}{2}\right]_{-1}^{2} \\ &=\left[\left(4-\frac{8}{3}+2\right)-\left(-2+\frac{1}{3}+\frac{1}{2}\right)\right] \\ &=\left[\frac{10}{3}+\frac{7}{6}\right] \\ &=\frac{27}{6} \\ &=\frac{9}{2} \text { sq.units } \end{aligned}

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