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  • Please solve RD Sharma Class 12 Chapter 20 Areas of Bounded Region Exercise 20.3 Question 5 Maths textbook solution.

Please solve RD Sharma Class 12 Chapter 20 Areas of Bounded Region Exercise 20.3 Question 5 Maths textbook solution.

Answers (1)

Answer:\frac{ab}{4}[\pi -2] Square units.

Hint:

Given:

\left\{(x, y): \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} \leq 1 \leq \frac{x}{a}+\frac{y}{b}\right\}

Solution:

\begin{aligned} &\text { Let } R=\left\{(x, y): \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} \leq 1 \leq \frac{x}{a}+\frac{y}{b}\right\} \\ &R_{1}=\left\{(x, y): \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} \leq 1\right\} \\ &\text { And } R_{2}=\left\{(x, y): 1 \leq \frac{x}{a}+\frac{y}{b}\right\} \\ &\text { Then } R=R_{1} \cap R_{2} \end{aligned}

considered \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 ,this represent an ellipse ,symmetrical about both axis and cutting x-axis at A(a,0)and A`(-a,0) and y-axis at B(0,b),B`(0,-b)

R_1=\left (\frac{x^2}{a^2}+\frac{y^2}{b^2}\leq 1 \right ) Represents the area inside the ellipse

\frac{x}{a}+\frac{x}{b}=1 Represent a straight line cutting x-axis at A(a,0) and y- axis at B(0,b)

R_2=\left (\frac{x}{a}+\frac{x}{b}\geq 1 \right ) Represent the area above the straight line

R=R_1\cap R_2 Represent the smaller shaded arera bounded by the line and the ellips in the shaded region, consider a vertical strip with length=\left | y_2-y_1 \right | and width=dx such that P(x,y_2)  lies on ellipse and (x,y_1) lies on the straight line area of approximating rectangle=\left | y_2-y_1 \right |dx

The approximating rectangle move from x = 0 to x = a 

\therefore Area of the shaded region=\int_{0}^{a}\left|y_{2}-y_{1}\right| d x=\int_{0}^{a}\left(y_{2}-y_{1}\right) d x \ldots \ldots \ldots \ldots .\left[A s, y_{1}\left|y_{2}-y_{1}\right|=y_{2}-y_{1}\right]

\begin{aligned} &A=\int_{0}^{a}\left(\frac{b}{a} \sqrt{a^{2}-x^{2}}-\frac{b}{a}(a-x)\right) d x \\ &A=\int a\left(\frac{b}{a} \sqrt{a^{2}-x^{2}}\right) d x-\int_{0} a \frac{b}{a}(a-x) d x \\ &A=\frac{b}{a}\left[\left\{\frac{x}{a} \sqrt{\left.\left.a^{2}-x^{2}+\frac{1}{2} a^{2} \sin ^{-1}\left(\frac{x}{a}\right)\right\}\right]_{0}^{a}-\frac{b}{a}\left[a x-\frac{x^{2}}{2}\right]_{0}^{a}}\right.\right. \\ &A=\frac{b}{a}\left[\frac{1}{2} a^{2} \times \frac{\pi}{2}-\frac{a^{2}}{2}\right] \\ &A=\frac{a b}{2}\left[\frac{\pi}{2}-1\right] \\ &A=\frac{a b}{4}[\pi-2] \text { Square units } \end{aligned}

 

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