Basic Integration Problem

1. Jan 10, 2012

IntegrateMe

Function given:

F(b) = ∫e-x2dx from 0 to b

I'm asked to estimate F(1), F(2), and F(3) using a left-sum with 3 subdivisions.

So, I guess Δx would be 1, then, so it doesn't really matter for the purposes of solving this problem. However, this is as far as I've gotten, and I haven't really been able to make much progress.

Can anyone suggest another step I can take to come closer to the solution?

Thank you!

2. Jan 10, 2012

cragar

so you want to estimate the area under this curve with 3 rectangles. So what is the width of the rectangle. And how tall is the rectangle?

3. Jan 10, 2012

IntegrateMe

The width of the rectangles would simply be 1, so they don't really matter in our calculations (i.e. we can simply add the heights together). I think the height of each rectangle would be e-x2 evaluated at 0, 1, and 2, but I'm unsure. Am I on the right track?

4. Jan 10, 2012

cragar

yes your on the right track. the value of the function at an x would be the height.

5. Jan 10, 2012

IntegrateMe

So,

F(1) = e-12
F(2) = F(1) + e-22
F(3) = F(2) + e-32

?

6. Jan 10, 2012

HACR

and should multiply each by b/3 to get the area since it's an integral.so $$\frac{b}{3}(e^{-1}+e^{-4}+e^{-9})=F(3)~F(b)$$

7. Jan 10, 2012

chapstic

F(1) means b = 1.

so you need to find ∫e-x2dx from 0 to 1.

left hand estimate with 3 divisions. the width of the rectangles does matter.

for F(1) you need to divide the space between 0 and 1 into 3 sections. I will start you off: 0, $\frac{1}{3}$,.....(obviously there are 2 more in order to divide into 3 equal sections).

provided that makes sense, you can then move on to find the height of each rectangle:
the height is going to be found by plugging in your LEFT-HAND values of x you found when you divided the graph into 3 parts.

since I told you 0 is one of your x-values, the height for 0 will be:

e-02 = 1

the resulting area is going to be the (width of the rectangles) x (the sum of the heights).

Last edited: Jan 10, 2012