I need help with these 3 questions

[DaAzNJRiCh]

1) Create an integral for (1^2+2^2+3^2+4^2+...+n^2)/(n^3)

2) Find the limit, as x approaches 5, of (2x/(x-5)) x the integral of [sin(t)]/t dt.
I think this has something to do with as f(a)-f(x)/(a-x)=f'(a) or something like that.

3) Evaluate the integral of square root of (b^2-x^2) db from the interval (b/square root of 2) to b. I need to show the graph without using a calculator or at least show where it should be.

You don't have to answer all questions but 1 would be nice.

 

[pka]

For #1.
For the function \(\displaystyle f(x) = x^2\) for the integral \(\displaystyle \int\limits_0^1 {f(x)dx}\) use the approximating sum \(\displaystyle \sum\limits_{k = 1}^n {f(x_k )\Delta x} ,\quad \Delta x = \frac{1}{n},\quad x_k = \frac{k}{n}\quad k = 1,2...,n\)

 

[stapel]

2) I'm sorry, but I don't know what you mean by "the limit of this, the integral of that". Are you taking a limit? Or finding an integral? (I've never heard of a limit as x approaches a value of an integral in another variable, is why I ask.)

3) Do you mean that you need to integrate sqrt[b² - x²]dx over the interval [b/sqrt[2], b]? (I'm not familiar with any meaning to "from the interval this to that", is why I ask.) Are you given a value of "b"? (How else are you supposed to show a graph of whatever you're doing?)

When you reply, please include the exact text of the exercises, and their instructions. Thank you.

Eliz.

 

[galactus]

Is this what you mean?:

#2. \(\displaystyle \L\\\lim_{x\to\5}\frac{2x}{x-5}\int\frac{sin(t)}{t}dt\)

At first glance, appears to be undefined.

#3: \(\displaystyle \L\\\int_{\frac{b}{\sqrt{2}}}^{b}\sqrt{b^{2}-x^{2}}db\)
Are you sure you don't mean dx?. This has a dependent limit.

 

[DaAzNJRiCh]

Yea

For the first one, I need it to be an integral of something from one point to another.

FOr the third one it is dx but how do I do it?

 

[galactus]

Re: Yea

For the first one, I need it to be an integral of something from one point to another.

What are the limits of integration?. If you don't know, I sure don't. :roll:

For the third one it is dx, but how do I do it?

You can use trig substitution. Let \(\displaystyle x=bsin({\theta}) \;\ and \;\ dx=bcos({\theta})d{\theta})\)

\(\displaystyle \L\\\int_{\frac{b}{\sqrt{2}}}^{b}\sqrt{b^{2}-b^{2}sin^{2}({\theta})}bcos({\theta})d{\theta}\)

\(\displaystyle \L\\\int_{\frac{b}{\sqrt{2}}}^{b}\sqrt{b^{2}(1-sin^{2}({\theta}))}bcos({\theta})d{\theta}\)

Now, remember your trig identities and finish up?.

Also, don't forget your new limits of integration.

\(\displaystyle \frac{b}{\sqrt{2}}=bsin({\theta}) \;\ and \;\ b=bsin({\theta})\)

Solve for \(\displaystyle {\theta}\)

 

[pka]

Re: Yea

For the first one, I need it to be an integral of something from one point to another.

Did you see the first response?
I gave you the exact answer: an integral from 0 to 1.

 

[galactus]

pka, I think they mean the first one in my first post, which is actually the 2nd problem.