# Math Help - please check integral problem

1. ## please check integral problem

basin has a form of cylinder with height of 0,5 meters and background radius of 0,3 meters. calculate a work that is needed to evacuate the water from the basin.

my solution: area of background = $Pi \cdot R^2$
as water has density of 1000 and $F=m \cdot g$ then
$A=m \cdot g \cdot x=882 \cdot Pi \cdot x$ and integrating the last expression from 0 to 0,5 i get 346,36 but an answer given for this problem is 346,54.

basin has a form of half of a sphere with radius 0,1 meters. calculate a work needed to evacuate water from it.

my solution: are of circle that is x meters under the center is
$Pi \cdot r^2 = Pi(R^2 - x^2) = Pi(0,01 - x^2)$
$A=m \cdot g \cdot h =Pi(0,01 - x^2) \cdot 10^4 \cdot x$
and as i integrate it from 0 to 0,1 i get 0,79, but the answer given is 1,54.

2. For the first one, I did the example myself and used g = 9.8 and got your value. Using a value of g = 9.805, I got 346.537, which rounded gives the desired result. However, I looked it up and got a value of 9.806 from Gravity of Earth - Wikipedia, the free encyclopedia (third paragraph).

If you take 9.789 at the equator, you get 345.97
And if you take 9.832 at the poles, you get 347.492

So I guess you should expect a small variation in your answer, depending on the exact constants used.

For the second part, it is somewhat ambiguous. Which way is the hemisphere facing?

3. ok, thanks for the first one. it is good if this variation in answer is caused by constants. the main thing i wanted to check whether i have applied the integral correctly.
for the second i thought the same way, but only difference would be that area of background is function of x.

4. No problem. I think you have applied the integral correctly. For the second case, judging by your picture, I think you've done the integration correctly. I did the calculation too and again used 9.805. I got 0.770083, which is 1/2 of 1.540166, which rounded to 3 sig figs is 1.54. It's possible that the answer in the book is off by a factor of 2 itself!

5. thanks alot. hope that it is a wrong answer in the book