MATH 116 ACTIVITY 3: Area between curves and the definite integral
WHY: Area between curves gives a nice “practice” application of the definite integral (some setup, but not too complex)
but also serves as a model of the total effect of a difference over time – total change in a population will be the
difference between the effect of birth rate and death rate, for example [so that total change would be represented
by the area between the birth curve and the death curve )
LEARNING OBJECTIVES:
1. Further understanding of the integral as a summing mechanism (adding up area)
2. Be able to set up the integrals necessary for calculation of areas.
3. Be able to calculate integrals using the fundamental theorem of calculus or approximate with a sum (as appropriate)
CRITERIA:
1. Success in completing the exercises.
2. Success in working as a team and in filling the team roles.
RESOURCES:
1. Your text - especially section 7.6
2. Your class notes from the last two days
3. The team role desk markers (handed out in class for use during the semester)
4. 40 minutes
PLAN:
1. Select roles, if you have not already done so, and decide how you will carry out steps 2 and 3
2. Read through the model and complete the exercises given here - be sure all members of the team understand and agree
with all the results in the recorder's report.
3. Assess the team's work and roles performances and prepare the Reflector's and Recorder's reports including team grade .
MODEL:
We can use the definite integral to calculate the area between two graphs in much the same way we use it to calculate the
area under the graph of a positive function. As before, this gives a representation of the total change - but this time we
allow for two rates of change interacting. We still want to multiply the base by height to get area – but this time the height
depends on two graphs (one at the bottom of the area, one at the top) rather than just one.
the rule is that the area between two graphs, from x = a to x = b, is given by
€
(y(top) − y(bottom))dx
a
b
∫
- if we set
up rectangles for approximation, the base for each is x and the height is (top – bottom).
If the graphs do not cross, this is straightforward – we use
€
f
top
(x) − f
bottom
(x)dx
a
b
∫
If the graphs cross, then the formula for y(top) sometimes comes from one function, sometimes from the other, and we
have to split up the integral using the
€
f(x)dx =
a
b
∫
f(x)dx +
a
c
∫
f(x)dx
c
b
∫
property of the definite integral.
Usually we need to sketch the graphs (take advantage of our calculator] to see whether the graphs cross and which graph is
on top where.
Example 1: Area between the graphs of
€
f1(x) = x2− 1
and
€
f2(x) = 2x+1
between x = 0 and x = 2.
