flamingtext_com_1242228210_9682.gif
flamingtext_com_1242228399_15647.gif




external image twilight-9.jpg


Definition:

The process of finding the area under a particular curve by dividing the area into many tiny rectangles, adding up the heights of individual rectangles, and then multiplying the sum by their common width.



Formulas:

calculus_formulas.jpg




Video:





Steps:


1. Sketch the region
2. Establish whether f(x) is increasing/decreasing
- If increasing: Left rectangular approximation (LRAM) will be the LOWER sum (less than actual area) and Right rectangular approximation (RRAM) will
be the UPPER sum (greater than actual area)
- If decreasing: LRAM will be UPPER sum and RRAM will be LOWER sum
3. Determine the number of rectangles or trapezoids (n)
4. Find delta x
- Width of each rectangle
- Height of each trapezoid
5. Use formulas to solve for the estimated Area



Examples:


1.)

example1.jpg




1.jpg





Graph made from http://cs.jsu.edu/mcis/faculty/leathrum/Mathlets/riemann.html#instr




2.)
2.jpg
example2.jpg


Graph made from http://cs.jsu.edu/mcis/faculty/leathrum/Mathlets/riemann.html#instr



example3.jpg3.)



NIGRAPH3.jpg



Graph made from http://cs.jsu.edu/mcis/faculty/leathrum/Mathlets/riemann.html#instr



4.)

external image moz-screenshot-19.jpg4.jpg

example4.jpg



5.)

real_life_example2.jpg
real_life_example3.jpg

http://www.calculus-help.com/probs1998/problem15.html


Practice Links:


Practice right, left, and midpoint approximations
Left, right and trapezoidal riemann sums
Upper and lower riemann sums