# Math Help - Finding the volume of a solid (disk/wash/shell. I get the answer wrong consistently.

1. ## Finding the volume of a solid with cross sections

Find the volume of the following solid using the method of your choice:

The solid whose base is the region bounded by y=x2 and the line y=1 and whose cross sections perpendicular to the base and parallel to the x-axis are semicircles.

It says to use V= ∫ba A(y) dy.
But I don't even know what area i'm supposed to find to use.

I am struggling with this section of calculus as a whole. I've watched tons of youtube videos and examples, but I still mess up if they get even slightly complicated. So if you can explain the reasoning as to why you made choices, it would be really helpful to my understanding. If not, then steps are fine. Thank you so much.

2. ## Re: Finding the volume of a solid (disk/wash/shell. I get the answer wrong consistent

Originally Posted by tanagholat
Find the volume of the following solid using the method of your choice:

The solid whose base is the region bounded by y=x2 and the line y=1 and whose cross sections perpendicular to the base and parallel to the x-axis are semicircles.

It says to use V= ∫ba A(y) dy.
But I don't even know what area i'm supposed to find to use.

I am struggling with this section of calculus as a whole. I've watched tons of youtube videos and examples, but I still mess up if they get even slightly complicated. So if you can explain the reasoning as to why you made choices, it would be really helpful to my understanding. If not, then steps are fine. Thank you so much.
I do not agree with that "answer"

diameter of a semicircular cross-section, $d = 1-x^2$

$r = \frac{1-x^2}{2}$

area of a cross section, $A(x) = \frac{\pi r^2}{2} = \frac{\pi(1-x^2)^2}{8}$

using symmetry across the y-axis ...

$V = \frac{\pi}{4} \int_0^1 (1-x^2)^2 \, dx$

3. ## Re: Finding the volume of a solid (disk/wash/shell. I get the answer wrong consistent

I went back and checked the answer that the mymathlab program said and it still says π/4.

I will wait and see what others say. Thank you again

4. ## Re: Finding the volume of a solid (disk/wash/shell. I get the answer wrong consistent

Originally Posted by tanagholat

I went back and checked the answer that the mymathlab program said and it still says π/4.

I will wait and see what others say. Thank you again
fine, you can wait.

for the record, $V = \frac{2\pi}{15}$

5. ## Re: Finding the volume of a solid (disk/wash/shell. I get the answer wrong consistent

Originally Posted by skeeter
I do not agree with that "answer"

diameter of a semicircular cross-section, $d = 1-x^2$
I think you're misreading the question. The cross-sections parallel to the x-axis are semicircles. The radius of each cross-section, in terms of $y,$ is $r = \sqrt{y}.$ Integrating with respect to $y,$ we have

$V = \int_0^1\frac12\pi r^2\,dy$

$= \frac\pi2\int_0^1\left(\sqrt y\right)^2\,dy$

$= \frac\pi2\int_0^1y\,dy$

$= \frac\pi4.$

6. ## Re: Finding the volume of a solid (disk/wash/shell. I get the answer wrong consistent

I just watched a video explaining cross sections with semi circles and after looking back at the way you solved this problem, it all just 'clicked' and made sense. Thank you very much to the both of you!

7. ## Re: Finding the volume of a solid (disk/wash/shell. I get the answer wrong consistent

Originally Posted by Reckoner
I think you're misreading the question. The cross-sections parallel to the x-axis are semicircles.
you are correct ... I saw the word "perpendicular" in the original post and thought perpendicular to the x-axis.

mea culpa.