3

3. Using the function/graph, calculate the area of each rectangle, and compute the sum that approximates
the area under the curve.
Introduction. Suppose we are interested in computing integrals of the following function
f(z) = ksin (rz).
within the interval (0,1), but this time definite integrals. In general, we as mathematicians, statisticians,
and practicing scientists are very interested in this kind of problem. This time around, we need to choose
the mumber & such that 1
[ ksin(az)dz = 1
0
We might not be ready to perform the definite imegration just yet, it is, after all, a definite integral with
u-substitution, something we haven't quite gotten to in class. But, we can chip away at the problem.
Suppose that & = §. In the first few questions of this assignment, using n = 4 rectangles, you will
fy
calculate a left-hand estimate for { 5 sin(rz)de.
(7
1. Draw a picture of the function f(z), which serves as the integrand, on the interval [0,1]. (Make a . : : . .
sketch, thinking about what sin(1z) iteelf looks like, and bow the coefficient in front $ influences the 4. Since the integrand is continuows, we the following template, a result of the Fundamental Theorem of
graph. Calculate a few key values to get started.). If you get stuck, you can trace from a calculator as, exact value,
screen, but try to think through it using the information suggested above. ty 1
[fostenae = ( it
‘a
Fill in the blank with the simplest antiderivative of f(z) = § sin(srr) and finish the calculation in the
space below.
2. Make a new copy of your sketch and draw the a = 4 left-hand rectangles. Is it an obvious overestimate
or underestimate? If so, which? if pot, why not?
5. Suppose we only wanted to integrate from z = 0 to x = 0.5, would left-hand rectangles be an overes-
timate or underestimate in this case? What. if anything, is different now compared to the earlier case
(ie., question 2)?