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  • Provide solution for RD Sharma Maths Class 12 Chapter 20 Areas of Bounded Region Exercise 20.3 question 16.

Provide solution for RD Sharma Maths Class 12 Chapter 20 Areas of Bounded Region Exercise 20.3 question 16.

Answers (1)

Answer:

\frac{16}{3}\; \text{ab sq. units} .

Hint:

Use basic concepts.

Given:

y^2=4ax and x^2=4by .

Solution: To find area enclosed by

y^2=4ax                ..........(1)

x^2=4by                .........(2)

Equation (1) represents a parabola with vertex (0,0) and axis as x-axis,

Equation (2) represents a parabola with vertex (0,0), and axis as y-axis,

Point of intersection of parabolas are (0, 0) and (4a^{\frac{1}{3}} b^{\frac{2}{3}} , 4a^{\frac{2}{3}}b^{\frac{1}{3}})

 

A rough sketch is given as:

The shaded region is required area  and it is sliced into rectangle of width=\Delta _x  and length(y_2-y_1)

Area of rectangle=(y_1-y_2)\Delta _x

This approximation rectangle slides from x=0 to, x= 4a^{\frac{1}{3}}b^{\frac{2}{3}} , so

Required area = Region OQAPO

\begin{aligned} &=\int_{0}^{4 a^{\frac{1}{4} b^{2}} 3}\left(y_{1}-y_{2}\right) d x \\ &=\int_{0}^{4 a^{\frac{1}{a} b^{2}}}\left(2 \sqrt{a} \cdot \sqrt{x}-\frac{x^{2}}{4 b}\right) d x \\ &=\left[2 \sqrt{a} \cdot \frac{2}{3} x \sqrt{x}-\frac{x^{3}}{12 b}\right]_{0}^{4 a^{\frac{1}{3} b^{2}}} \\ &=\frac{32 \sqrt{a}}{3} \cdot a^{\frac{1}{3}} b^{\frac{2}{3}} a^{\frac{1}{6}} b^{\frac{1}{3}} \\ &=\frac{64 a b^{2}}{12 b} \\ &=\frac{16}{3} a b \text { sq.units } \end{aligned}

Hence the required area is \frac{16}{3}\; \text{ab sq. units} .

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