1. Approximating Areas Under Curves

Good morning!

I'm having a bit of a hard time wrapping my mind around this concept. The part that seems to be troubling me the most is where Riemann Sums and sigma notation come in. I can't seem to find explanations online that help me better understand, so hopefully someone here could help?

The textbook that I am using is Briggs and Cochran's Calculus, section 5.1 specifically. For anybody that has this book, I have tried to follow example #5 as it seemed like it would be the most helpful. I have no clue what exactly they've done. The problem asks you to evaluate the left, right, and midpoint Riemann sums of f(x)= x^3 + 1 between a=0 and b=2 using n=50 subintervals. I can't figure out what to do once I have my Δx. The book just leaves me lost.

Thanks!

2. Re: Approximating Areas Under Curves

You'd have to be very lucky if a member of the forum has the exact same book.

3. Re: Approximating Areas Under Curves

Sorry, haha. I provided the question from the example I was referring to in that paragraph. But here's a specific problem I'd love to see explained:

For the given value of n, use sigma notation to write the left, right, and midpoint Riemann sums. Then evaluate each sum using a calculator.
f(x)=x^2 - 1, for [2,7]; n=75

I'm not exactly sure what to do once I have my delta x. I think one of the biggest problems I'm having is with calculating the Riemann sums for large values of n.

4. Re: Approximating Areas Under Curves

$\sum_{i=0} ^{n-1} \Delta x f(x_l + i \Delta x)$ for left,

$\sum_{i=0} ^{n-1} \Delta x f(x_l + (i + 1/2) \Delta x)$ for middle, and

$\sum _{i=1} ^{n} \Delta x f(x_l + i \Delta x)$ for right?

$x_l$ means the lower boundary of the interval, in this case, 2; $f(x)$ the function you want to compute your Riemann sums for.

Hope this helps,

Bart