Problem Set 16 – Area Functions
Problem Set 16 – Area Functions
16.1In this section, we will consider how the area under a curve changes as the interval changes.
1)The figures below show the graph of .
a)Shade the area under the graph of the function for each interval specified below.
Shade from 0 to 1Shade from 0 to 2Shade from 0 to 3
b)Fill in the table. Plot the points (right endpoint, area) on the grid.
Right endpoint / 1 / 2 / 3 / 4 / 5 / xArea /
c) What is the rule for the “area function?”
2)Draw the graph of on the grid below.
a)Compute the areas under the graph of f using 0 as the left endpoint of the interval, and fill in the table below. Plot the (right endpoint, area) points on the grid below.
Right endpoint / 1 / 2 / 3 / 4 / 5 / xArea
b) Find a rule for the “area function”.
c) Use this area function to compute the area under the graph of fon the interval [0, 8.2]
Problem Set 16 – Area Functions
3)The figures below show the graph of .
a)Shade the area under the graph of the function for each interval specified below.
Shade from 0 to 1Shade from 0 to 2Shade from 0 to 3
b)Fill in the table. Plot the points on the grid.
Right endpoint / 1 / 2 / 3 / 4 / 5 / xArea
c)Find a rule for the “area function”.
d) Use this area function to compute the area under the graph of fon the interval [0, 10].
Problem Set 16 – Area Functions
4)The left endpoint of the interval doesn’t need to be 0.
a)Shade the area under the graph of the function for each interval specified below.
Shade from 1 to 1Shade from 1 to 2Shade from 1 to 3
b)Fill in the table. Plot the points on the grid.
Right endpoint / 1 / 2 / 3 / 4 / 5 / xArea
c)What is the relationship between the graphs you drew in 3b and 4b?
16.2We find the area function by looking at area under the curve. Thus the area function can be written as an integral.
5)Below we see the graph of . On the left we see the area represented by . Write the integral that represents the area shown on the right.
In integral notation, the xis called a “dummy variable” or the “variable of integration”. In fact, the integral can be written as or . They all represent the same number. We wish to use this idea to write the area under a curve as a function of the right endpoint of an interval.
Notice that the area function you wrote for the second graph above is a function of t because the area depends on where t is on the x-axis. We write
.
Alternatively, one could write . It is considered poor notation to write , since x is serving two roles: the dummy variable and the right endpoint of the interval.
6)Go back to problems 1-4 and write the area as a function of the right endpoint t using integral notation.