Area of bounded Regions

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Question

$The area bounded by the loop of 4 y^{2} = x^{2} (4 - x^{2}) is$

Moderate

Solution

$\begin{array}{l} 4 y^{2} = x^{2} (4 - x^{2}) - - - (i) \\ \Rightarrow y = \frac{1}{2} \sqrt{x^{2} (4 - x^{2})} \Rightarrow y = \frac{x}{2} \sqrt{(4 - x^{2})} \\ ∴ Area (A) = 4 \times \int_{0}^{2} \frac{2 x}{2 \times 2} \sqrt{(4 - x^{2})} d x \\ Put (4 - x^{2}) = t \Rightarrow - 2 x d x = d t \\ ∴ A = \frac{- 4}{4} \int_{4}^{0} \sqrt{t} d t = \int_{0}^{4} \sqrt{t} d t = {[\frac{t^{3 / 2}}{3 / 2}]}_{0}^{4} = \frac{2}{3} \times [\sqrt{64} - 0] \\ \Rightarrow A = \frac{16}{3} sq.units \end{array}$

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Area of bounded Regions

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The area bounded by the loop of 4y2=x24−x2 is

4y2=x24−x2−−−(i)⇒y=12x24−x2⇒y=x24−x2∴ Area (A)=4×∫02 2x2×24−x2dx Put 4−x2=t⇒−2xdx=dt∴A=−44∫40 tdt=∫04 tdt=t3/23/204=23×[64−0]⇒A=163 sq.units

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$The area bounded by the loop of 4 y^{2} = x^{2} (4 - x^{2}) is$