Calculate volume of solid obtained by rotating region bounded by $y=x^2$, line $y= −2$, from $x=0$ to $x=2$ around line $y=−2$.

Question

Problem : Calculate the volume of the solid obtained by rotating the region bounded by the function $y=x^2$, the line $y= −2$, from $x=0$ to $x=2$ around the line $y=−2$.

My Attempt

Can someone guide more clearly? Thanks

Answer 1

Make a slice parallel to the $y-$axis of thickness $dx$. Think of it as a cylindrical disk, with height (thickness) $dx$, centered on line $y=-2$. Then the radius of this disk is $y+2$. Thus the volume of this disk is $\pi(y+2)^2\, dx$. Now we stack all such slices to compute the total volume. So $$\text{Volume} = \int_{\color{red}{x=0}}^{\color{red}{x=2}} \pi (y+2)^2 \, d\color{red}{x} =\pi\int_0^2(x^2+2)^2 \, dx.$$ Hopefully you can complete it now.

Answer 2

Now, Let's think it like this. If we are bounded above by x-axis instead of $y=x^2$ then we would have a perfect cylinder with radius=2, height=2 integral of this cylinder can be written as

$$\pi \int_0^2 (y^2)dx $$ where y is constant which is 2.

Now returning to your question we can see that y is not constant it depends on x. and y can be writtten as $y=2+x^2$ so putting this into above integral gives us

$$ \pi \int_0^2 (2+x^2)^2dx$$ you can handle the rest.