Can you figure out the rest?Estimating F(b) Using Left-Sum with 3 Subdivisions

[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!

 

[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?

 

[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?

 

[cragar]

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

 

[IntegrateMe]

So,

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

?

 

[HACR]

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

 

[chapstic]

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.

Thank you!

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).