Given the integral e^x dx between a & b exists, evaluate it using the formula 1+r+r^2

**gummy_ratz** · November 1st 2010, 02:01 PM

Given the integral e^x dx between a & b exists, evaluate it using the formula 1+r+r^2...r^n = 1 - r^(n+1)/(1-r), where r doesn't = 1.

I tried, using (delta)x = (b-a)/n and xi = a + i(delta)x, but when I get to taking the limit I get stuck. I'm not even sure if that's what I'm suppose to do... but I'm not sure of any other way to go about it. Please help!

**Drexel28** · November 1st 2010, 02:04 PM

Originally Posted by gummy_ratz

Given the integral e^x dx between a & b exists, evaluate it using the formula 1+r+r^2...r^n = 1 - r^(n+1)/(1-r), where r doesn't = 1.

I tried, using (delta)x = (b-a)/n and xi = a + i(delta)x, but when I get to taking the limit I get stuck. I'm not even sure if that's what I'm suppose to do... but I'm not sure of any other way to go about it. Please help!

What level is this? What definition of integral are you using?

**Also sprach Zarathustra** · November 1st 2010, 02:19 PM

I think he (OP) meant using RIEMANN sums.

Integration of e^x using riemann sums? - Yahoo! Answers

**gummy_ratz** · November 1st 2010, 04:37 PM

The course is Real Analysis. And we didn't evaluate any Riemann sums like that, but I'm guessing that must be what they want me to do. The question just says:
"Given the integral between a & b e^x dx exists, evaluate it using the formula: 1 + r + r^2 + ... + r^n = (1-r^(n+1))/(1-r).

I think that yahoo link might be something like what I need to do, but mine is just generally between a and b.

And the definition of the integral I have is for every E>0 there's a d>0 : ||P|| < d -> |sigma - L| < E ... but I'm not really sure if I would apply that to this question?

Do you guys know how I should go about solving this?

**HallsofIvy** · November 2nd 2010, 05:31 AM

Divide the interval from a to b into n equal parts, each of length $\Delta x= \frac{b- a}{n}$ . The value of the function $f(x)= e^x$ at the left end of each sub-interval, is $e^a$ , $e^{a+ \Delta x}= e^a(e^{\Delta x}$ , $e^{a+ 2\Delta x}= e^a(e^{2\Delta x})= e^a(e^{\Delta x})^2$ , etc.

In other words, the integral is the limit of $e^a\Delta x \left(1+ (e^{\Delta x})+ (e^{\Delta x})^2+ \cdot\cdot\cdot+ (e^{\Delta x})^n\right)$ . Do you see what "r" must be?

**gummy_ratz** · November 2nd 2010, 06:43 PM

Okay great, thanks. Yeahh i get that. But where I run into trouble is when I try to take the limit.

I keep having 1 - e^(delta x) on the bottom, which = 1-1 = 0 as n -> oo, and so I don't know how to get rid of that?

**Drexel28** · November 2nd 2010, 07:34 PM

Originally Posted by gummy_ratz

Okay great, thanks. Yeahh i get that. But where I run into trouble is when I try to take the limit.

I keep having 1 - e^(delta x) on the bottom, which = 1-1 = 0 as n -> oo, and so I don't know how to get rid of that?

Try multiplying top and bottom by $latex e^{\Delta x}$

**gummy_ratz** · November 2nd 2010, 08:11 PM

Originally Posted by Drexel28

Try multiplying top and bottom by $latex e^{\Delta x}$

But I think e^(delta x) = 1 as x-> 00, since

delta x = (b-a)/n = 0 as x-> 00 and e^0 = 1

so that would just be multiplying the top and bottom by 1?

I have:

lim x->00 = e^(a+deltax)*(1- e^(deltax*n+deltax))/(1-e^delax)

but then even when I factor e^deltax out of the top and bottom, I'm still left with
((1/e^deltax) - 1) on the bottom.

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