Right Endpoint Integration Example

[Jason76]

Estimate area, 4 right endpoints and rectangles. Is this an overestimate or underestimate?

\(\displaystyle f(x) = 4 \cos x\) on interval \(\displaystyle [0,\dfrac{\pi}{2}]\) given \(\displaystyle [a, b]\)

\(\displaystyle \sum\limits_{i=4}^n \Delta x [f(a + i \Delta x)]\)

\(\displaystyle \Delta x = \dfrac{b - a}{n}\)

\(\displaystyle \Delta x = \dfrac{\dfrac{\pi}{2} - 0}{4} = \dfrac{\pi}{8}\)

\(\displaystyle n = 4\)

\(\displaystyle \sum\limits_{i=4}^n (\dfrac{\pi}{8}) [[f(0 +(1)(\dfrac{\pi}{8})] + [f(0 +(2)(\dfrac{\pi}{8})] + [f(0 +(3)(\dfrac{\pi}{8})] + [f(0 +(4)(\dfrac{\pi}{8})]]\)

\(\displaystyle \sum\limits_{i=4}^n (\dfrac{\pi}{8}) [[f(0 + (\dfrac{\pi}{8})] + [f(0 + (\dfrac{2\pi}{8})] + [f(0 + (\dfrac{3\pi}{8})] + [f(0 + (\dfrac{4\pi}{8})]]\)

\(\displaystyle \sum\limits_{i=4}^n (\dfrac{\pi}{8}) [[f(0 + (\dfrac{\pi}{8})] + [f(0 + (\dfrac{\pi}{4})] + [f(0 + (\dfrac{3\pi}{8})] + [f(0 + (\dfrac{\pi}{2})]]\)

\(\displaystyle \sum\limits_{i=4}^n (\dfrac{\pi}{8}) [[f(\dfrac{\pi}{8})] + [f(\dfrac{\pi}{4})] + [f(\dfrac{3\pi}{8})] + [f(\dfrac{\pi}{2})]]\)

\(\displaystyle \sum\limits_{i=4}^n (\dfrac{\pi}{8}) [[4 \cos (\dfrac{\pi}{8})] + [4 \cos (\dfrac{\pi}{4})] + [4 \cos (\dfrac{3\pi}{8})] + [4 \cos (\dfrac{\pi}{2})]]\)

\(\displaystyle \sum\limits_{i=4}^n (\dfrac{\pi}{8}) [[4(.9239)] + [4(.7071)] + [4(.3827)] + [4(0)]] \)

\(\displaystyle \sum\limits_{i=4}^n (\dfrac{\pi}{8}) [[3.6956] + [2.8284] + [1.5308] + [0] \)

\(\displaystyle \sum\limits_{i=4}^n (.3925) [[3.6956] + [2.8284] + [1.5308] + [0] \)

\(\displaystyle \sum\limits_{i=4}^n [.56933] + [1.1101] + [.6008] + [0] \)

 

[stapel]

What is your question?

 

[Jason76]

The main problem seems to exist in rounding. The idea is to use the full number until the very end, and then round. If you try to round too early, then it comes out wrong. But otherwise, the procedure in these problems is right, with the exception of putting the summation symbol on the last line.

 

[DrPhil]

The main problem seems to exist in rounding. The idea is to use the full number until the very end, and then round. If you try to round too early, then it comes out wrong. But otherwise, the procedure in these problems is right, with the exception of putting the summation symbol on the last line.

The summations should ALL be from i=1 to 4, NOT i=4,4.

You have expanded the sum, so you shouldn't have a summation sign anyhow.

The problem is NOT roundoff, but rather that you are using the right edge of each \(\displaystyle \Delta x\) interval as a representation of the value of the function over the whole interval. Look at the function: is it consistently increasing or decreasing? what will the affect be on the calculated area?

Integrate the same function and compare to the right-edge sum. Were you correct in your estimate of whether the sum would be larger or smaller than the exact integral?