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LUCKY_DIMON [66]
2 years ago
9

Find the formula for the area of the shaded region in the figure. x² = 4py ہے

Mathematics
1 answer:
Alexandra [31]2 years ago
5 0

The formula for the area of the shaded region in the figure.

x² = 4py is mathematically given as

A=\frac{8}{3} \sqrt{p h} \cdot h

<h3>What is the formula for the area of the shaded region in the figure?</h3>

Parameters

$x^{2}=4 p y$

$y=h$

\therefore x^{2}=4 p h \Rightarrow x=2 \sqrt{p h} \\

Generally, the equation for the Area is mathematically given as

& A=2 \int_{0}^{2 \sqrt{p h}}\left(h-\frac{x^{2}}{4 p}\right) d x \\\\& A=2\left(h x-\frac{x^{3}}{12 p}\right]_{0}^{2 \sqrt{p h}} \\\\& A=2\left[\left[h 2 \sqrt{p h}-\frac{1}{12 p} \cdot(2 \sqrt{p h})^{3}-0\right]\right. \\

&A=2\left[2 h^{3 / 2} \sqrt{p}-\frac{8(\sqrt{p})^{3}(\sqrt{h})^{3}}{12 p}-0\right] \\\\&A=2\left[2 h^{k / 2} \sqrt{p}-\frac{2}{3} \sqrt{p} \cdot h^{3 / 2}\right] \\\\&A=2 \times \frac{4 \sqrt{p} \cdot h^{3 / 2}}{3} \\\\&A=\frac{8}{3} \cdot \sqrt{p} \cdot \sqrt{h} \cdot h \\\\&A=\frac{8}{3} \sqrt{p h} \cdot h

A=\frac{8}{3} \sqrt{p h} \cdot h

In conclusion,  the equation for the Area is

   

A=\frac{8}{3} \sqrt{p h} \cdot h

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