Improper integral question help

[Destiny of Pi]

Hey guys, I don't really get the concept of improper integral and here are two questions I couldn't solve. If any of you can show me the steps of how to solve it, that will be appreciated. Thank you!

 

I think for question 9, a, it means 1/(1+x^2)

Edited January 6, 2018 by Destiny of Pi

[Destiny of Pi]

And also, this question doesn't make sense to me either...

what does it mean to find an upper and lower limit?

 

Edited January 6, 2018 by Destiny of Pi

[Nomenclature]

Treat an improper integral as you would a normal integral. That is, evaluate it.

Evaluating 9(a), I recognize (1/a^2+x^2) as an inverse arctan trig integral (arcsin, arccos, and arctan are derivatives you should know. I believe they'll be on the formula sheet during the external, but being familiar with them will help you). So we have [arctan(x)] on the boundary (infinity, -infinity). Either by thinking about it logically (as the slope of the radius on the unit circle approaches infinity i.e. a vertical line, what will the angle cast out be?) or by plugging the graph or arctan(x) into a calculator you should see that as x approaches infinity, arctan(x) approaches pi/2. Using the same methods, we see that as x approaches -infinity, arctan(x) approaches -pi/2. So subbing in the bounds to our integral, we know it equals pi/2 - (-pi/2) = 2pi/2 = pi. The integral converges to pi.

Evaluating 9(b) I use integration by parts, as u-substitution will not work. (Fun tip: to remember the formula uv - integral(du), I like the mnemonic device "ultraviolet voodoo"). Let x be u as its derivative, 1, will result in a solvable integral. So u = x, du = 1, dv = e^(-2x), v = e^(-2x)/-2. From there you should be able to solve (If you can't, write the equation and see how it solved here). The answer is (2x+1)e^(-2x)/-4. Solving for when x = 0, we get -1/4. When x = -infinity, the answer is -infinity * infinity / 4, which can be written as -infinity and so we can now look at the whole integral. -1/4 - (infinity) = -infinity, thus the integral diverges.

 

Edit: Always get help and check your work kids. 

Edited January 6, 2018 by Nomenclature

[Nomenclature]

Question 10 again has some simple, straightforward integration, and it has the by parts method that I just demonstrated. Please try to solve that using the above first and if you get stuck, then feel free to ask more questions.

[Nomenclature]

Question 6 is just a Riemann sum. Plot the points on a piece of paper. Than connect the points with straightforward lines. Logically, what should you do to get the maximum area under the curve? (remember, the curve you just drew is an approximation using only a few points. The actual curves area may be greater or less than that of your curve.

 

Hopefully, you arrived at this conclusion on your own. Do think about it for a minute before reading on. You should split the curve into three parallelograms and to get the maximum area you should treat them as three rectangles where the width is the difference between the two x-values and height is that of the greater of the two y-value. To get the minimum area, do this but use the lesser of the two y-values.

 

The answer I got is:

Spoiler

21 sq. units for the maximum; 11 sq. units for the minimum

 

Edited January 7, 2018 by Nomenclature

[kw0573]

9a. Use formula booklet to look up the integral (arctan x) and get pi.

9b. f(-inf) approaches -infinity so intergal does not converge because integrand is not bounded.

10. udv = uv - vdu

If you look up Gamma function, 10b 10c should match the motivation to define such a function (analytic contiuation of the factorial). 

6. Assuming 4 is the max value (question is impossible if at say x=7.5 f(x) is huge number like 9000). Some intuition is needed. If rectangles for Riemann sum goes above the positive curve, you overestimates the integral; if the rectangles are within the curve, you are underestimating.

Edited January 6, 2018 by kw0573

[Destiny of Pi]

10 hours ago, Nomenclature said:

Question 6 is just a Riemann sum. Plot the points on a piece of paper. Than connect the points with straightforward lines. Logically, what should you do to get the maximum area under the curve? (remember, the curve you just drew is an approximation using only a few points. The actual curves area may be greater or less than that of your curve.

 

Hopefully, you arrived at this conclusion on your own. Do think about it for a minute before reading on. You should split the curve into three parallelograms and to get the maximum area you should treat them as three rectangles where the width is the difference between the two x-values and height is that of the greater of the two y-value. To get the minimum area, do this but use the lesser of the two y-values.

 

The answer I got is:

  Hide contents

25 sq. units for the maximum; 11 sq. units for the minimum

 

2

Thanks for the explanation!!

But for the max., I got 21

3*3+2*4+1*4=21

[Destiny of Pi]

For question 10 part c, I still don't really know how to evaluate r(n) in terms of n. Doesn't the formula already do that?

[kw0573]

6. 21, seconded. 

10c. It want you to say Gamma = (n-1)!, which can be derived from parts a and b. Basically the integral form is function of both n and x, but they want it as function of just n. If you take more math courses, you will study more functions of multiple input variables as well as functions that use an integral as definition.

[Nomenclature]

9 hours ago, Destiny of Pi said:

Thanks for the explanation!!

But for the max., I got 21

3*3+2*4+1*4=21

Yep. You're right. Sorry about that. 

[Destiny of Pi]

I also came across this question

it's about the sum and product of roots, but I'm kinda stuck on this...

 

[kw0573]

1 hour ago, Destiny of Pi said:

I also came across this question

it's about the sum and product of roots, but I'm kinda stuck on this...

 

Use variables to denote the four roots and expand the factored form to find coefficient of x term.