18.01 Section, December 8, 2015
Section leader: Eva Belmont (ebelmont@mit.edu, E18-401B)
.
Riemann sums
1. Estimate
R1
t
f (t)
0
f (t) dt, where all we know about f is the following:
0
0.8
0.2
1.1
0.4
1.2
0.6
1.0
0.8
0.7
1
0.6
Do this:
(a) using rectangles where the left endpoint is the function value;
(b) using rectangles where the right endpoint is the function value;
(c) using trapezoids.
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2. In this problem, you will estimate ln(2) based on the fact that
R
1
x
= ln x.
(a) What integral do you need to estimate, in order to estimate ln(2)?
(b) Come up with a lower bound for ln(2) by using four bars. Draw a picture.
(c) Come up with a upper bound for ln(2) by using four bars. Draw a picture.
3. Suppose Rf ≥ 0 is a function that is zero everywhere except at a finite number of points.
∞
What is −∞ f (x) dx? Justify your answer using Riemann sums.
4. Express
Rπ
0
x sin(x) dx as a limit of sums.
2