Monday Oct 30

Solve, write-up and turn in:

1. Consider the function f(x)=2x on the interval [0,b].

Set up a Riemann sum to calculate the area under f(x)=2x for x in [0,b] as follows.

A. Consider a partition of [0, b] made up of 4 intervals of equal lengths. What are x0,x1,x2,x3,x4?

a) On each interval [xk,x k+1] take the intermediate points to be the left points (so x*k=xk).

Sketch the four rectangles and write the Riemann sum (do not simplify!)

Is this Riemann sum less, more, or equal to the actual area? Justify your answer by looking at the picture.

b) Same questions when you choose x*k to be the right points (so x*k=x k+1).

c) Same questions when you choose x*k to be the mid-points (so x*k=(xk+x k+1)/2).

B. Consider a partition of [0, b] made up of n intervals of equal lengths. Write x0,x1,x2,...xk,...xn.

On each interval take the intermediate points to be the left points (so x*k=xk).

(i) Write the Riemann sum.

(ii) Calculate the sum.

(iii) Calculate the limit as n-> infinity and show that this limit gives the area indeed.

(There was an y=ax floating around...the function should be y=2x.)

Solve, write-up and turn in: 2. Consider the function f(x)=a x^2 on the interval [0,b] (the one we did in class today).

Write a Riemann sum associated to n equal intervals, and with intermediate points being midpoints.

Calculate the limit as n goes to infinity of this Riemann sum.

What should this limit be? Explain!

Sec. 6.6: 3, 5, 9, 11, 13, 16, 20, 37, 42

Solve, write-up and turn in: Sec. 6.7: 1a, d (Caution: sketch the graph and note that part of the function is negative,

so the geometrical area of that part is "minus the definite integral"),

3, 7, 9, 11, 13, 15a, 16 a,b

3. Calculate the area between the graphs of y=sin x and y=cos x for x in [0,pi].

Caution: sketch the two graph and note that cos x is bigger on one interval, and sin x is bigger on another interval.

Solve, write-up and turn in: Sec. 7.2: 20, 25, 29a, 31, Bonus: 33

Sec. 6.7: 14

Sec. 7.3: 1 c, f, 2, 11

Solve, write-up and turn in: Sec. 7.3: 7, 9, 15, 22, 25