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Expert: - 2/28/2009


Question
QUESTION: Dear Scott, I try to evaluate the following Beta function: f(t)= ∫ from 1 to 4 (x-1)^2(4-x)^3 dx. I have to substitute the integrand 1 to 0 and 4 to 1 to get the right answer, but I don't know how to do so. Can you give me a hint? Thanks in advance and best regards, Martin

ANSWER: The only way I know to do this is to multiply it out.

(x-1)²=x²-2x+1.

(4-x)^3 = 64 - 48x + 12x^2 - x^3, which you can reverse and get
-x^3 + 12x^2 - 48x + 64.

Mulitply this by x², by -2x, and by 1.
Add them up and get the result, then multiply the entire result by 4.

The best way to do it is just like a grade school multiplication
table.  Do the 1 first and write the equation below.
Do the -2x next, and write that right below the last one,
shifted a space to the right.  Shift it over just like you
use to shift the 10's over.  Then multiply by x², and shift
that over like the 100's use to be shifted over.

When all done, add up what's in the 1's column, the x's column,
the x² column, the x^3 column, and so on until the problem is
complete.  I don't know of any easier way to do it.


---------- FOLLOW-UP ----------

QUESTION: Hi Scott, thanks for your answer. But my problem is to evaluate the integral from 1 to 4 of the whole Beta function. And therefor I have to substitute the integrands to 0 and 1. Are you familiar with this? Thanks in advance, kind regards, Martin

Answer
To change the value on the integrands, you have to change the variable.

For example, the ∫(0 to 4)x² dx would be 4³/3 - 0³/3 = 64/3.
If we replace the x with t=x/4, then the limits would change,
but so would the integral.  We have to make note that dt = dx/4,
so dx = 4dt.

We could then transform the integral into  ∫(0 to 1)(4t)²4 dx.
Note that (4x)²4 is really 64x², so the integral would be still be
64/3.

So if the limits are changed from 0 to 4 to 0 to 1, the integral is also changed by the multiplication of 4.

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Scott A Wilson

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I can answer any question in general math, arithetic, discret math, algebra, box problems, geometry, filling a tank with water, trigonometry, pre-calculus, linear algebra, complex mathematics, probability, statistics, and most of anything else that relates to math. I can even tell you it takes me over 2,000 steps to go a mile, but is that relevant?

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Experience in the area; I have tutored people in the above areas of mathematics for almost two years in AllExperts.com. I have tutored people here and there in mathematics since before I received a BS degree almost 25 years ago. In just two more years, I received an MS degree as well, but more on that later. I tutored at OSU in the math center for all six years I was there. Most students offering assistance were juniors, seniors, or graduate students. I was allowed to tutor as a freshman. I tutored at Mathnasium for well over a year. I worked at The Boeing Company for over 5 years. I received an MS degreee in Mathematics from Oregon State Univeristy. The classes I took were over 100 hours of upper division credits in mathematical courses such as calculus, statistics, probabilty, linear algrebra, powers, linear regression, matrices, and more. I graduated with honors in both my BS and MS degrees. Past/Present Clients: College Students at Oregon State University, various math people since college, over 7,500 people on the PC from the US and rest the world.

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My master's paper was published in the OSU journal. The subject of it was Numerical Analysis used in shock waves and rarefaction fans. It dealt with discontinuities that arose over time. They were solved using the Leap Frog method. That method was used and improvements of it were shown. The improvements were by Enquist-Osher, Godunov, and Lax-Wendroff.

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Master of Science at OSU with high honors in mathematics. Bachelor of Science at OSU with high honors in mathematical sciences. This degree involved mathematics, statistics, and computer science. I also took sophmore level physics and chemistry while I was attending college. On the side I took raquetball, but that's still not relevant.

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I earned high honors in both my BS degree and MS degree from Oregon State. I was in near the top in most of my classes. In several classes in mathematics, I was first. In a class of over 100 students, I was always one of the first ones to complete the test. I graduated with well over 50 credits in upper division mathematics.

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