TABLE OF CONTENTS
Let f(x) = sin x and a = 0. Define
To find the value F(x), we integrate the sin function from 0 to x.
(a) To find F(π), we integrate sin from 0 to π:
This means we're accumulating the weighted area between sin t and the t-axis from 0 to π:
Evaluating the integral, we get
The value of F(π) is the weighted area between sin t and the horizontal axis from 0 to π, which is 2.
(b) To find we substitute in for x:
On the graph, we're accumulating the weighted area between sin t and the t-axis from 0 to .
The value 1 makes sense as an answer, because the weighted areas
cancel each other out:
We're left with one-half the area
Since we found that
it makes sense for
to be 1.
(c) To find we put in for x. The upper limit of integration is less than the lower limit of integration 0, but that's okay.
You can write the answer as
if you want.
On the graph, we're accumulating the weighted area between sin t and the t-axis from 0 to . This means we're integrating going left:
Since we're accumulating area below the axis, but going left instead of right, it makes sense to get a positive number for an answer.
Let . Find each value and represent each value using a graph of the function 2t.
The value F(2) is this area:
(b) Since we're integrating over an interval of length 0,
When we try to represent this on a graph, we get a line, which has no area:
Since we're integrating to the left, F(0) is the negative of this area:
(d) Since 2t is an odd function,
The areas above and below the t-axis on [-1,1] are the same:
The weighted area between 2t and the t-axis on [-1,1] is 0, so we're left with the area on [-2,1]. Since we're integrating to the left and this area is below the t-axis, we count this area positively: