ক্ষেত্রফল নির্ণয়

The area, in square units of the region bounded by the parabolas y2=4xy^{2}=4x and x2=4yx^{2}=4y is

হানি নাটস

The given parabolas are y2=4xy^{2}=4x and x2=4yx^{2}=4y

The point of intersection of x2=4yx^2=4y and y2=4xy^2=4x will be

x=4x=4 and x=0x=0

Then, the required area will be

04(4xx24)dx\overset{4}{\underset{0}{\displaystyle \int}}\left (\sqrt{4x} -\dfrac{x^2}{4} \right )dx

=[4×23x3/2x312]04=\left[\sqrt{4}\times\dfrac{2}{3}x^{3/2}-\dfrac{x^3}{12}\right]_{0}^{4}

=4×834312=\dfrac{4\times 8}{3}-\dfrac{4^3}{12}

=323163=\dfrac{32}{3}-\dfrac{16}{3}

=163squnits=\dfrac{16}{3}\:sq\:units

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