How do I do these Calculus Problems?

I need to make an integral that works for all of this...
(1^2+2^2+3^2+4^2+...+n^2)/(n^3)

I need to find the limit as x approaches 5 of this: (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.


Last one I need help on is 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.

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Originally Posted by DaAzNJRiCh

I need to make an integral that works for all of this...
(1^2+2^2+3^2+4^2+...+n^2)/(n^3)

I need to find the limit as x approaches 5 of this: (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.


Last one I need help on is 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.

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This is confusing.

What are you asking for in the first question? It looks to me like you want a closed form expression for the summation? ( (n+1)(2n+1)/(6n^2) )

The integral has nothing to do with x, so the limit does not exist. I'm having a hard time understanding why this question was asked because the answer is too simple.

I can give you a hint as to how to do the integral (sub b = x*sec(theta)), but what do you mean by a graph? The integral is a real number.

-Dan

Hello, DaAzNJRiCh!

Like Dan, I don't understand these problems . . . at all.

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I need to make an integral that works for all of this...
. . (1² + 2² + 3² + ... + n²)/n³

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Could you give us the original wording of the problem?

We can simplify that expression:
. . n(n + 1)(2n + 1)/6n³ .= .(n + 1)(2n + 1)/6n²

but I don't see how we can "make" an integral.

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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.

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First of all, the integral should have dx . . . right?
. . It appears that b is a constant.
. . . . . . . . . . . . . . .______
The graph of: .y .= .√b² - x² .is a semicircle with center (0,0) and radius b.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ._
And we want the area under the curve from b/√2 to b.

To make an integral use f(x)=x^2 on [0,1].
Using right-hand Riemann Sums with delta(x)=1/n .

1.I need it so that is an integral like the integral of f(x) from 1 point to another. Something like that.

2. I need to find the limit as x approaches 5 of this: (2x/(x-5)) times the integral of [sin(t)]/t dt.


3. And it is dx I think but I need to find the value as well.

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Originally Posted by DaAzNJRiCh

2. I need to find the limit as x approaches 5 of this: (2x/(x-5)) times the integral of [sin(t)]/t dt.

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By sin(t)/t you mean,
g(x)=INTEGRAL^x_1 sin(t)/t dt