1. Estimating With Finite Sums

can somebody explain to me how to do the following problem?

y = 2x-x^2 [0,2]

partition [0,2] into 4 subintervals and show the four rectangles that LRAM uses to approximate the area of the function. Compute the LRAM sum without a calculator.

thanks for any help

2. Originally Posted by turtle
can somebody explain to me how to do the following problem?

y = 2x-x^2 [0,2]

partition [0,2] into 4 subintervals and show the four rectangles that LRAM uses to approximate the area of the function. Compute the LRAM sum without a calculator.

thanks for any help
I assume you need to find,
$\displaystyle \int_0^2 2x-x^2dx$

The length of the interval is 2.
And there are 4 rectangles, so the width of each one is,
$\displaystyle \Delta x=\frac{2}{4}=.5$

Now form the sum,
$\displaystyle \sum_{k=1}^n f(a+k\Delta x)\Delta x$
But,
$\displaystyle \Delta x=.5$
$\displaystyle a=0$
$\displaystyle n=4$
$\displaystyle f(x)=2x-x^2$
(Note, $\displaystyle k$ is a running value. It has no specific value. It simply are all the integers between 1 and 4).
Thus, $\displaystyle f(a+k\Delta x)=f(.5k)$
Thus,
$\displaystyle \sum_{k=1}^4 f(.5k)\Delta x$
Thus,
$\displaystyle \sum_{k=1}^4 [2(.5)k-(.5k)^2](.5)$
Thus,
$\displaystyle \sum_{k=1}^4 (k-.25k^2)(.5)$
$\displaystyle (1-.25(1)^2)(.5)+(2-.25(2)^2)(.5)+(3-.25(3)^2)(.5)+(4-.25(4)^2)(.5)$