Limit definition of definite integrals

**fattydq** · February 6th 2009, 01:34 PM

In a practice quiz I'm working on for my calc class I'm asked to evaluate the given definite integral from 3 to 7 of (x^2-3x+5)dx using the limit definition of a definite integral. I know how to express an integral as a limit using the limit definition, and I know how to evaluate definite integrals usint areas, but I don't know how to EVALUATE a definite integral using the limit definition. What does this even mean exactly and how could I go about doing it?

**Mush** · February 6th 2009, 01:42 PM

Solutions to the Limit Definition of a Definite Integral

Problem 5 is especially relevant.

**Abu-Khalil** · February 6th 2009, 01:45 PM

You may define the partition $x_0<x_1<\ldots<x_{n-1}<x_n$ , where $x_k=3+\left(\frac{7-3}{n}\right)k\Rightarrow \Delta x_k=x_k-x_{k-1}=\left(\frac{7-3}{n}\right)$ so when $n\to\infty, \Delta x_k \to 0$ .

Finally, $\int_3^7x^2-3x+5dx=\lim_{n\to\infty}\frac{7-3}{n}\sum_{k=1}^nx_k^2-3x_k+5$ and calculate that sum isn't a problem knowing $\sum_{k=1}^nk=\frac{n(n+1)}{2}$ and $\sum_{k=1}^nk^2=\frac{n(n+1)(2n+1)}{6}$ .

**fattydq** · February 6th 2009, 01:47 PM

thanks guys I'll check that link out for sure

**fattydq** · February 6th 2009, 01:50 PM

Originally Posted by Abu-Khalil

You may define the partition $x_0<x_1<\ldots<x_{n-1}<x_n$ , where $x_k=3+\left(\frac{7-3}{n}\right)k\Rightarrow \Delta x_k=x_k-x_{k-1}=\left(\frac{7-3}{n}\right)$ so when $n\to\infty, \Delta x_k \to 0$ .

Finally, $\int_3^7x^2-3x+5dx=\lim_{n\to\infty}\frac{7-3}{n}\sum_{k=1}^nx_k^2-3x_k+5$ and calculate that sum isn't a problem knowing $\sum_{k=1}^nk=\frac{n(n+1)}{2}$ and $\sum_{k=1}^nk^2=\frac{n(n+1)(2n+1)}{6}$ .

I'm sorry that's a bit too jumbled for me I don't really understand any of what you said. What is x subscript k even representing here?

**fattydq** · February 8th 2009, 02:49 PM

I keep getting wrong answers. I just found the answer by using the integral function on my calculator but I can't seem to get it. Could someone walk me through this step by step?

Thread: Limit definition of definite integrals

LinkBack

Thread Tools

Display

Limit definition of definite integrals

Similar Math Help Forum Discussions

[SOLVED] Definite Integrals - Same Problem, 1st use Definite Integral, then FTC.

Calculate definite integral (4x^2) using definition as a limit of approximating sums

Finding a definite integral from other given definite integrals.

Limit of definite integrals

definite integrals, upper limit variable

Search Tags