Limit process to find area of regions

[lamaclass]

I'm unsure of how to set up these types of problems.

Here's a couple from my book:

Use the limit process to find the area of the region between the graph of the function and the x-axis over the given interval.

1. y=-2x+3, [0,1]

2. 64-x[sup:1skslo47]3[/sup:1skslo47], [1,4]

Also how do you determine if the left-endpoint or right-endpoint is better to use? Any help is greatly appreciated!

 

[chrisr]

1. Check whether or not the graph crosses the x axis in that interval (it doesn't).
You do this to ensure you calculate area using integration.
2. Is f(x) + or - from 0 to 1? (it's +). If it was -, you'd reverse the sign.

Then your limits of integration are 0 and 1,
so now you integrate the function from x=0 to x=1.
First, write the integral of 3-2x.
The constant of integration is not required when evaluating area.

You will be subtracting the integral evaluated with x=0 from the integral evaluated at x=1.
Basically, you will be subtracting zero.

Try that and see if you can progress to the second problem.

 

[BigGlenntheHeavy]

\(\displaystyle !) \ y \ = \ -2x+3, \ [0,1], \ always \ use \ the \ right\ \ endpoint \ unless \ instructed \ otherwise \ as \ it \ is \ the\)

\(\displaystyle \ easiest \ (less \ grunt \ work).\)

\(\displaystyle \Delta x \ = \ \frac{b-a}{n} \ = \ \frac{1-0}{n} \ = \ \frac{1}{n}.\)

\(\displaystyle Right \ endpoint \ = \ M_i \ = \ 0+i(1/n) \ = \ \frac{i}{n}\)

\(\displaystyle Hence, \ \lim_{n\to\infty}S(n) \ = \ \ \lim_{n\to\infty} \sum_{i=1}^{n}f(M_i)\Delta x\)

\(\displaystyle = \ \lim_{n\to\infty}\sum_{i=1}^{n}\bigg[-2\bigg(\frac{i}{n}\bigg)+3\bigg]\frac{1}{n}\)

\(\displaystyle = \ \lim_{n\to\infty} \frac{1}{n^{2}} \sum_{i=1}^{n}[-2i+3n}]\)

\(\displaystyle = \ \lim_{n\to\infty}\frac{-2}{n^{2}}\sum_{i=1}^{n}i+\lim_{n\to\infty}\frac{3n}{n^{2}}\sum_{i=1}^{n}1\)

\(\displaystyle = \ \lim_{n\to\infty}\frac{-2}{n^{2}}\bigg[\frac{n(n+1)}{2}\bigg]+3\)

\(\displaystyle = \ \lim_{n\to\infty}-\frac{n}{n}-\frac{1}{n}+3 \ = \ 2 \ QED\)

\(\displaystyle 2) \ \Delta x \ = \ \frac{3}{n} \ and \ M_i \ = \ 1+\frac{3i}{n}, \ I'll \ let \ you \ finish \ it.\)