Description
We stated in class that computing anti-derivatives from a specific point to another is an area
under the curve. We can approximate this with rectangles and triangles.
1 2 3 4 5
5
10
15
20
25
(2, 4)
(1, 1)
(3, 9)
(4, 16)
x
f(x)
For this problem, we can estimate the area under this curve from 1 to 4 by computing the
areas explicitly.
The first column can be approximated by the area 1
2
(1)(3) = 3
2
(for the triangle) and (1)(1) =
1 is the area. The total area is 5
2
.
The second column can be approximated by the area 1
2
(1)(5) = 5
2
(for the triangle) and
(1)(4) = 4. The total area 13
2
.
THe third column can be approximated by the area 1
2
(1)(7) = 7
2
(for the triangle) and
(1)(9) = 9. The total area 25
2
.
Putting these all together, we get the area 5
2 +
13
2 +
25
2 =
43
2
.
Some things to notice: first, the y-values are coming from the function x
2
. The area of a
rectangle is base times height. And the area of a triangle is half of base times height.
1. Estimate the area under 2x
for x-values between 1 and 4.
2. Estimate the area under x
3
for x-values between 1 and 3.