# Thread: Volume of an object

1. ## Volume of an object

A volume is described as follows:
1. The base is the region bounded by $x= -y^{2} + 8y + 21$ and $x=y^{2} - 18y + 93$;
2. Every cross section perpendicular to the y-axis is a semi-circle.
Find the volume.

So you would find the area by doing:

$\int_4^{9}(-y^{2} +8y + 21 -( y^{2} -18y + 93))dy$

Right? But then i don't understand how you would find the volume....

2. i am also not getting the second part of your question:every cross-section perpendicular to the y-axis is a semi-circle. the equations given result into 2 parabolas but will the area obtained by their intersection be a semi-circle???

3. No the area will look like the region where the 2 equations intersect. For the cross sections part, imagine the picture i attatched but instead of the rectangular part (the area), imagine the region where the 2 equations instersect in place of it. The end shape will look sortof like a canoe.

4. This problem is a bit tricky because you must find cross sections perpendicular to the y-axis (i.e., vertical), so you need to define the curve as an equation of the form y=f(x).

You can solve for them by using any method you might use to solve a quadratic equation (e.g., quadratic equation, completing the square). You will have to determine which of the two solutions you get is correct for the boundaries of the given region. That is, for example, if you use the quadratic equation, you will get two solutions (one with + and one with - before the radical). You only need to use one of them in each case.

Once you have the equations, you can then find cross sections of the region that correspond to the diameter of the semicircle. So, if the diameter is d(x), then the area of a cross section of the volume is:

$A=\frac{1}{2} \pi r^2 = \frac{1}{2} \pi \left( \frac{d(x)}{2} \right)^2 = \frac{\pi}{8} \left(d(x)\right)^2$

So, integrating these cross sections gives the total volume:

$V = \frac{\pi}{8} \int_{x_1}^{x_2} \left(d(x)\right)^2 dx$

Where $x_1,x_2$ are the endpoints of the region.

Is this enough explanation for you to solve the problem?