# Math Help - Help Please need Clarfication Integral Volumes.!

1. ## Help Please need Clarfication Integral Volumes.!

The question is;
Find the volume of the solid whose base is the region bounded by
y = e^x, y = 1, x = 2 and x = 3 ,

where cross-sections perpendicular to the
x-axis are semicircles.

The part of this equation I do not understand is the set up, am not sure how all y=1 comes into play? and how i should revolve this graph to get a solid of revoultion? basically dont know were to start? If anyone please help me with a starting point thank you .

2. Originally Posted by zangestu888
The question is;
Find the volume of the solid whose base is the region bounded by
y = e^x, y = 1, x = 2 and x = 3 ,

where cross-sections perpendicular to the
x-axis are semicircles.

The part of this equation I do not understand is the set up, am not sure how all y=1 comes into play? and how i should revolve this graph to get a solid of revoultion? basically dont know were to start? If anyone please help me with a starting point thank you .
this is not a solid of revolution ... it is a solid formed by similar cross-sections.

$V = \int_a^b A(x) \, dx$

in this case, each similar cross section is a semicircle. the diameter of a semicircular cross-section is $d = e^x - 1$ , so, the radius of a semicircular cross-section is $r = \frac{e^x-1}{2}$

therefore, the representative cross sectional area as a function of x is

$A(x) = \frac{1}{2} \pi \left(\frac{e^x-1}{2}\right)^2$

limits of integration will be from x = 2 to x = 3.

you have all the information you need ... set up the volume integral and evaluate.

3. ## Thanks :)

So $d = e^x - 1$, the $-1$ comes from the fact that the region has the line y=1 correct, that was the part that was really the most confusing I just cant picture all of this sometimes, is thier a trick lol thanks! much appreciated

4. Originally Posted by zangestu888
The question is;
Find the volume of the solid whose base is the region bounded by
y = e^x, y = 1, x = 2 and x = 3 ,

where cross-sections perpendicular to the
x-axis are semicircles.

The part of this equation I do not understand is the set up, am not sure how all y=1 comes into play? and how i should revolve this graph to get a solid of revoultion? basically dont know were to start? If anyone please help me with a starting point thank you .

First you need to draw the graph of what this looks like. The region bounded by these curves is above $y=1$, below $y=e^x$ and between $x=1$ & $x=2$. This is the area that you will be finding the volume of.

I assume you have learned how to use estimating rectangles? Draw your estimating rectangle from $y =e^x$ to $y=1$. This is the section that if you moved it back and forth would create semi-circles perpendicular to the x-axis, it would NOT be a full revolution! .