Question

Find out volume of solid obtained by rotating about x +axis the area of the parabola y^(2)=4ax cut off by its latus rectum.​

Answer 1

$\underline{\textbf{Given:}}$

$\textsf{Curve is}\;\mathsf{y^2=4ax}$

$\underline{\textbf{To find:}}$

$\textsf{Volume of the solid obtained by rotating about x-axis the area}$

$\textsf{of the parabola cut off by its latus rectum}$

$\underline{\textbf{Solution:}}$

$\textsf{Equation of latus rectum of the parabola}\;\mathsf{y^2=4ax\;is\;x=a}$

$\textsf{So, limits are x=0 and x=a}$

$\underline{\textbf{Volume of the solid formed}}$

$\mathsf{=\displaystyle\int\limits^b_a\,\pi\;y^2\;dx}$

$\mathsf{=\displaystyle\pi\int\limits^a_0\,4ax\;dx}$

$\mathsf{=\displaystyle\,4a\pi\int\limits^a_0\,x\;dx}$

$\mathsf{=4a\pi\left(\dfrac{x^2}{2}\right)\limits^a_0}$

$\mathsf{=\dfrac{4a\pi}{2}\left(x^2\right)\limits^a_0}$

$\mathsf{=2a\pi(a^2-0)}$

$\mathsf{=2a^3\pi\;cubic\;units}$

$\underline{\textbf{Find more:}}$

Find area of the circle x^2+y^2=36 by using definite integration

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