Question

Integral problem.

Answer 1

I'm not sure if my notation is correct

Answer 2

It keeps saying it is incorrect....

Answer 3

The Riemann sum is defined as:
\[ for\left[ a,b \right] \in \pi \iff \int\limits_{a}^{b}f(x)dx=\lim_{n \rightarrow \infty}\sum_{i=0}^{n}f(a+\frac{ b-a }{ n }i)(\frac{ b-a }{ n })\]

Answer 4

"n" is the number of parts you divide the region, they can be rectangles or hexagons, I personally prefer the rectangles.
But since these are two parts, we don't have to take the limit, leaving us with only:
\[A \approx \sum_{i=0}^{i=2} f(a+\frac{ b-a }{ 2 }i)(\frac{ b-a }{ 2 })\]
Let's first understand what \(\frac{ b-a }{ n }\) means, it is but the "division" of the region where "b" and "a" are the limits of the sum so we can begin by calculating it:
\[\Delta x=\frac{ b-a }{ n } \iff \Delta x=\frac{ 1-0 }{ 2 } \iff \Delta x=\frac{ 1 }{ 2 }\]
now we can plot it:
\[A \approx \sum_{i=0}^{i=2} f(a+\frac{ 1 }{ 2 }i)(\frac{ 1 }{ 2 })\]
"a" as I said, is but the lower limit of the integral, and fortunately it's zero:
\[A \approx \sum_{i=0}^{i=2}f(\frac{ 1 }{ 2 }i)(\frac{ 1 }{ 2 })\]
But look, we can't just apply it for now, we have to find what \(f(\frac{ 1 }{ 2 }i)\) is, and it is just 1/2i evaluated in the function e^x:
\[f(x)=e^x \iff f(\frac{ 1 }{ 2 }i)=e ^{\frac{ 1 }{ 2 }i}\]
Thus:
\[A \approx \sum_{i=0}^{i=2}(e ^{\frac{ 1 }{ 2 }i})(\frac{ 1 }{ 2 })\]
Now, this summatory isn't so compley, so we can express it as the sum of two terms, we can skip the i=0 term, because it yields no area value, if you look at the definition:
\[\sum_{i=0}^{i=2}(e ^{\frac{ 1 }{ 2 }i})(\frac{ 1 }{ 2 })=(e ^{\frac{ 1 }{ 2 }})(\frac{ 1 }{ 2 })+(e ^{\frac{ 1 }{ 2 }2})(\frac{ 1 }{ 2 })\]
Therefore:
\[A \approx (e ^{\frac{ 1 }{ 2 }})(\frac{ 1 }{ 2 })+(e ^{\frac{ 1 }{ 2 }2})(\frac{ 1 }{ 2 })\]
And this should be simple enough for you to handle.

Answer 5

so \[A \approx ((e^.5)/2)+e/2\] ?

Answer 6

@Owlcoffee

Answer 7

Correct.
if you simplify the operations you will obtain indeed:
\[A \approx e ^{\frac{ 1 }{ 2 }}(\frac{ 1 }{ 2 })+e(\frac{ 1 }{ 2 })\]
\[A \approx \frac{ e ^{\frac{ 1 }{ 2 }} }{ 2 }+\frac{ e }{ 2 }\]

Answer 8

I recommend you use fractions in order to make things more exact and simpler, there are some tutorials on fractions here in Openstudy, you can ask @pooja195 to send you the link.

Answer 9

okay...it says the first half is correct but the second half is not...

Answer 10

Hmm, it is not correct to plot e(0.5)?

Answer 11

The program recognize (e^(1/2))/2 as an answer but not e/2

Answer 12

This is what a left Riemann sum looks like.