So okay this is another find the volume of the cylinder type of question.
Step 1 Recognise which axis is the region rotated on.
Step 2 Know the volume of the cone and note that the x-axis represents the height using the concept of the cone through the rotation about the x-axis in
![](https://latex.codecogs.com/png.latex?\bg_white y=x)
Step 3 Write in
![](https://latex.codecogs.com/png.latex?\bg_white \pi\int_{0}^{1}y^{2}dx)
because the boundaries are 0 and 1 which is the height thus explaining this part
![](https://latex.codecogs.com/png.latex?\bg_white \int_{0}^{1})
Step 4 Write in
![](https://latex.codecogs.com/png.latex?\bg_white \pi\int_{0}^{1}x^{2}\left(1-x^{2}\right)^{2}dx)
Step 5 Expand
![](https://latex.codecogs.com/png.latex?\bg_white \left(1-x^{2}\right)^{2}=1-2x^{2}+x^{4})
Step 6 Multiply by
![](https://latex.codecogs.com/png.latex?\bg_white x^{2})
Step 7 Integrate
![](https://latex.codecogs.com/png.latex?\bg_white \pi\int_{0}^{1}x^{2}-2x^{4}+x^{6}dx)
Step 8 Solve
![](https://latex.codecogs.com/png.latex?\bg_white \frac{1}{3}-\frac{2}{5}+\frac{1}{7})
by completing the calculation.
What you will see here
@Farhanthestudent005 is that you need to know what the volume of the cylinder is but with a minuscule height because at the end of the day when you rotate a graph along either the x or y-axis or z-axis if you are in 2nd-year uni or above you are inevitably going to have circles and the area of the circle is
![](https://latex.codecogs.com/png.latex?\bg_white \pi{r^{2}})
so then to find the volume of the cylinder you have to have
![](https://latex.codecogs.com/png.latex?\bg_white \pi{r^{2}}h)
or to recognise the pattern it will be
![](https://latex.codecogs.com/png.latex?\bg_white \pi\int_{lower bound}^{upper bound}\left(x \cup y\right)^{2}d\left(Rotation along the ... axis\right))