# Use areas to evaluate ∫

1. Feb 25, 2013

### dillon131222

The Question

Let f(x) = |x|. use areas to evaluate ∫(-1,x)f(t)dt for all x. use this to show that d/dx∫(0,x)f(t)dt = f(x)

not sure hot to evaluate the integral using area when i dont know what f(t) is...

Last edited: Feb 25, 2013
2. Feb 25, 2013

### Dick

f(x)=|x|. So f(t)=|t|.

3. Feb 25, 2013

### epenguin

You know what f is.

You know what mathematical symbolism is.

4. Feb 25, 2013

### Karnage1993

As to your next question, I'm not sure what you mean by "use areas", but I recommend that you draw out what f(x) looks like. Then, see if you can spot an elementary shape the area under -1 to 0 looks like and the same with the area under 0 to x.

5. Feb 25, 2013

### dillon131222

oh.. ya thats kinda obvious now that you point it out :P thanks :)

ya thats basically what using the area is :P just didnt clue in to what f(t) was :P

6. Feb 25, 2013

### dillon131222

so heres my attempt:

http://img692.imageshack.us/img692/3284/graphed.png [Broken]
with f(x) = |x| so f(t) = |t| graphed above, and the area from -1 to x would be

(1/2)t2 -1/2 = ∫(-1,x)f(t)dt, so

d/dx(∫(0,x)f(t)dt) = f(x)

d/dx(1/2x2) = |x|

x = |x|

that seem correct?

Last edited by a moderator: May 6, 2017
7. Feb 25, 2013

### Karnage1993

Yes, it's correct, but I have to be picky in how you showed it. You should start with the LHS of what you want to show, ie, d/dx∫(0,x)f(t)dt, and simplify it to f(x). Like this:

LHS
= d/dx∫(0,x)f(t)dt
= d/dx((1/2)x^2)
= x
= |x|.....[since x >= 0]
= f(x), which is what we wanted to show. □

Last edited: Feb 25, 2013