Expert: Paul Klarreich - 5/22/2008

Question
QUESTION: Hello Paul,
I need some help with this problem
Let f(x) be a continuous function that is defined for all real numbers x and that has the following properties.

integral 1 to 3 [f(x)dx] =16
integral 1 to 5 [f(x)dx] =44

Evaluate integral 3 to 5 [3f(x)+2]dx
   Given that f(x) is a linear function of x, find an expression for f(x)

what I did so far : I broke up the integral and evaluated the integral 3 to 5 expression to be 88
then where do I go from there?
PS: I have attached my work
Thankyou

ANSWER: Questioner:   sami
Category:  Calculus
Private:  No
 
Subject:  integrals

..............................
Hi Sami,

You wrote:
...........................
I need some help with this problem
Let f(x) have the following properties.

integral 1 to 3 [f(x)dx] =16
integral 1 to 5 [f(x)dx] =44

>> OK, now -- I think that

integral 3 to 5 [] = integral 1 to 5 - integral 1 to 3

= 44 - 16 = 28
.............................

Evaluate integral 3 to 5 [3f(x)+2]dx
Given that f(x) is a linear function of x, find an expression for f(x)

what I did so far : I broke up the integral and evaluated the integral 3 to 5 expression to be 88

>> That does not look right.  UNLESS, of course, you mean the entire expression.

then where do I go from there?
PS: I have attached my work
Thankyou

================================
Evaluate integral 3 to 5 [3f(x)+2]dx

{5
| [3f(x)+2]dx =
}3


{5         {5   
3| f(x)dx + | 2 dx =
}3         }3    


3(28) + 2x[3 to 5]

3(28) + 2[5 - 3]

84 + 2(2) = 88

Yes, that looks right.


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

QUESTION: Hi Paul , thank you for verifying my work.
The part I needed help in completing i s
"Given that f(x) is a linear function of x, find an expression for f(x)"
How do I do this?
Thankyou


Answer
QUESTION: Hi Paul , thank you for verifying my work.
The part I needed help in completing i s
"Given that f(x) is a linear function of x, find an expression for f(x)"
How do I do this?
Thankyou
................................
Oh, yes -- sorry. I didn't see that.

Write f(x) = ax + b and put it into the integral equations:

integral (1 to 3) [(ax + b)dx] =16, then work out the integral:

ax^2/2 + bx [1..3] = 9a/2 + 3b - (a/2 + b) = 16
9a/2 + 3b - a/2 - b = 16

4a + 2b = 16  << first equation.
...............
integral (1 to 5) [(ax + b)dx] =44,

ax^2/2 + bx [1..5] = 25a/2 + 5b - (a/2 + b) = 44

25a/2 + 5b - a/2 - b = 44

12a + 4b = 44 << second equation.

Now just solve those for a and b and you have your answer.