Activity the fundamental theorem of calculus

A. Approximation with Riemann Sums The area under a curve can be approximated through the use of Riemann sums: Let Ln = Sum of n rectangles using the left-hand x-coordinate of each interval to find the height of the rectangles (LRAM) Let Rn = Sum of n rectangles using the right-hand x-coord of each interval to find the height of the rectangles (RRAM) Let Mn = Sum of n rectangles using the midpoint x-value of each interval to find the height of the rectangles (MRAM)

a) If all the intervals are the same width and there are Let f (x) = 9 – x2 on [0, 3] a) If all the intervals are the same width and there are six rectangles, what is the value of xk? Use this value to compute the following. Draw a sketch of the function along with the indicated rectangles. Use your calculators with the Lists in the STAT section to compute the following Riemann sums. (Commands for the TI-83/84 are shown.) List1 = x-coordinate for each rectangle List2 = y-coordinate for each rectangle (using formula in title bar) List3 = width of interval List4 = Area of each rectangle (using formula in title bar) From the home screen 2nd LIST, MATH, SUM(L4) will sum the values in List4 0.5

Left-hand Rule LRAM A = 20.125 L6 = 0 .5 1 1.5 2 2.5 3 Right-hand Rule RRAM A = 15.625 R6 = 0 .5 1 1.5 2 2.5 3

Midpoint Rule MRAM A = 18.0625 M6 = .25 1.25 2.25 .75 1.75 2.75

b) Suppose xk had a width equal 0. 5 b) Suppose xk had a width equal 0.5. Choose a random value for xk in each interval and compute the area of each Riemann rectangle. Find the total of the rectangles. Random x-coordinate Mrs. Fox = 18.22 Class Average = .4 1.4 2.4 .6 1.6 2.6

You will do #2 for homework! c) Choose six randomly sized intervals and choose a point at random in each interval. Compute the area of the Riemann rectangle and find the total area. Random rectangles Mrs. Fox = 18.144 Class Average = 1 .3 .5 .3 .5 .4 You will do #2 for homework! .5 1.45 2.35 1.15 2.05 2.8

The definite integral is simply the limit of the Riemann sum as the width of the interval gets smaller and smaller as indicated in the formula below. The fnInt option on your calculator uses Riemann sums with x = 0.0001. It computes the area of each rectangle and gives the total of all rectangles from x = a to x = b.

Limits went “backwards” Sketch a quick graph and shade the indicated region of each of the following functions. Use fnInt to approximate the value of each definite integral. 3) 4) 5) MATH 9 = 2.667 = –2.667 = 2.667 Limits went “backwards” Why is the answer to problem 4 a negative number?

Draw a sketch for each problem Draw a sketch for each problem. (Make sure your calculator is set to RADIAN mode for these problems!) Use fnInt to approximate the value of each definite integral. 6) 1 7) –2

8) 9) –2

10) 2 11) 2

homework WS Problem #2