Improper Integral and Force on a Dam

[ineedhelpnow]

my final is tomorrow and my instructor gave a list of questions that will be similar to the ones on the final exam and i want to see how they should be done properly. I've been working on other problems but i can't get past these ones. thanks

$\int_{-\infty}^{0} \ \frac{dx}{(2+x) \sqrt{x}}$

a graph of $y=\frac{x^2}{4}$ is given with intersection point $(4,4)$. find the total force of the dam using coordinate system at the bottom of the dam. (i don't even know what this question means)

 

[ineedhelpnow]

there's an example here that's similar to the second question but it doesn't really make sense?? it's on the last page.

View attachment 2876

 

[Dethrone]

I'm not at the level of math that you are on, but I can help you on the first improper integral. If you make the substitution \(\displaystyle x = u^2\), the integral will simplify into a form that you can easily take its anti-derivative. Then you can proceed as you would normally with a normal improper integral, taking into consideration that it has a discontinuity at x = -2, and x = 0.

 

[ineedhelpnow]

appreciate the help Rido :)for that one i thought i needed to integrate it t to 0 and then take the limit as t goes to -$\infty$ i realized my mistake because i hadn't noticed that's undefined at -2 (thank you for pointing that out) so i guess i have to integrate from -$\infty$ to -2. and then from -2 to 0.

 

[MarkFL]

You might find this thread helpful for the question regarding hydrostatic forces:

http://mathhelpboards.com/questions-other-sites-52/ns-questions-yahoo-answers-regarding-hydrostatic-forces-6150.html

 

[topsquark]

$\int_{-\infty}^{0} \ \frac{dx}{(2+x) \sqrt{x}}$

I'm not quite sure why no one has made this comment...The integrand does not exist for negative x because of the square root. There is no integration to do here. Or am I missing something obvious?

-Dan

 

[Dethrone]

I noticed that too, because if you tried to integrate, you'd have to sub in negative infinity into arctan(sqrt(x/2)), but I didn't want to comment because I wasn't sure.

 

[ineedhelpnow]

when i was integrating it i got some weird answers but i tried to ignore it and kept going on. :D

 

[ineedhelpnow]

for the interval on the dam (in the example), the bottom of the dam was at the point 0 on the y-axis but why did the interval go upto 25?