Understanding Integration: Solving a Tricky Calculus Problem

[flying_young]

Dear all,

Sorry for the silly integration question. I haven't done calculus in ages and my memory has got really rusty! ='(

The integral question is as follows:

\int\ (X_1-\frac{X_2}{2})^2(\frac{1}{X_2}) dX_1

Sorry, I could not figure out how to insert the limits on my intergral. It is from 0 to X_2.

My answer is:

= \int\ (\frac{X_1^2}{X_2}-X_1 + {\frac{X^2}{4}) dX_1

Which, after integrating, becomes

(\frac{X_2^2}{3}-1/2X_2^2 + X_2^2/8 ]

In the end, I get -X2^2/24, but the answer in the book is X2^2/12. What am I doing wrong? Thanks in advance for your help!

 

[NoMoreExams]

Dear all,

Sorry for the silly integration question. I haven't done calculus in ages and my memory has got really rusty! ='(

The integral question is as follows:

\int\ (X_1-\frac{X_2}{2})^2(\frac{1}{X_2}) dX_1

Sorry, I could not figure out how to insert the limits on my intergral. It is from 0 to X_2.

My answer is:

= \int\ (\frac{X_1^2}{X_2}-X_1 + {\frac{X^2}{4}) dX_1

Which, after integrating, becomes

(\frac{X_2^2}{3}-1/2X_2^2 + X_2^2/8 ]

In the end, I get -X2^2/24, but the answer in the book is X2^2/12. What am I doing wrong? Thanks in advance for your help!

So let me rewrite what you have:

\int_{0}^{X_2} \left( X_1 - \frac{X_2}{2}\right)^2\frac{1}{X_2} dX_1

Note that you can take \frac{1}{X_2} outside of the integral since it's a constant w.r.t. X_1 (as well as taking out the \frac{1}{4})

So expanding out the square we get:

\frac{1}{4X_2} \int_{0}^{X_2} 4X_{1}^{2} - 4X_{1}X_{2} + X_{2}^{2} dX_{1} = \frac{1}{4X_2} \left( \frac{4X_{1}^{3}}{3} - 2X_{1}^{2}X_{2} + X_{1}X_{2}^{2} \right)_{0}^{X_2}

And finally plugging in the limits we get

\frac{1}{4X_2} \left(\frac{4}{3}X_{2}^{3} - 2X_{2}^{3} + X_{2}^{3} \right) = \frac{1}{12} X_{2}^{2}

 

[NoMoreExams]

Just a side note the notation that you can with limits is

\int_{a}^{b} f(x) dx or \sum_{i=0}^{n} p(x)

 

[HallsofIvy]

NoMoreExams, that was nicely done but after making the point that X₂ was a constant and that you can multiply the square out (or make the substitution v= X₁- X₂/2) it would have been better to let flying young finish the problem himself.

 

[NoMoreExams]

Good point, I will do so in the future.

 

[flying_young]

Thank you, NoMoreExams! I finally figured out what I did wrong - a silly integration mistake! Thanks again! You have been an immense help.

 

[flying_young]

I know very much to treat X2 as a constant, but if NoMoreExams hasn't written out his steps, I wouldn't have figured out where I have gone wrong in the midst of calculations. I do understand from your stance that simply giving out the answer would do one no good. Perhaps in the future, I will be sure to write out my steps and have the pros here point out what I have done wrong. Thank you!

NoMoreExams, that was nicely done but after making the point that X₂ was a constant and that you can multiply the square out (or make the substitution v= X₁- X₂/2) it would have been better to let flying young finish the problem himself.