Question

Use the formula y=pi integral(x^2) to derive the formula of volume of sphere. Anyone

Answer 1

\[y=\pi \int\limits_{a}^{b}x^2\]

Answer 2

We just have to utilise this formula. Values of y and x can be anything

Answer 3

Note: The formula is used to calculate the volume inside the line or curve when it is rotated 360 degrees with respect to x axis

Answer 4

I don't know what you mean by \(y\) and \(x\), but it sounds like you want to derive the volume of a sphere using integration with either shells or disks.

Consider the upper semicircle defined by the function \(y=\sqrt{1-x^2}\) (so the radius is \(1\), for simplicity). Then revolving once about the \(x\)-axis, you have
\[V=\pi\int_{-1}^1\left(\sqrt{1-x^2}\right)^2\,\mathrm{d}x=2\pi\int_0^1(1-x^2)\,\mathrm{d}x\]which is easy enough to compute and determine that the volume would in this case be \(\dfrac{4}{3}\pi\).