How to Evaluate Integrals Using Areas: A Scientific Approach

[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 don't know what f(t) is...

 

[Dick]

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∫f(t)dt = f(x)

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

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

 

[epenguin]

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∫f(t)dt = f(x)

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

You know what f is.

You know what mathematical symbolism is.

 

[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.

 

[dillon131222]

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

oh.. you that's kinda obvious now that you point it out :P thanks :)

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.

ya that's basically what using the area is :P just didnt clue into what f(t) was :P

 

[dillon131222]

so here's my attempt:

http://img692.imageshack.us/img692/3284/graphed.png
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?

 

[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. □