http://www.ck12.org Chapter 14. Concepts of Calculus
The area of the first box is 2 times the height of the function evaluated at 3:
2 ·(^12 · 3 − 2 )= 3 − 4 =− 1
Because this box is under thex-axis, its area is negative.
The area for each of the rest of the boxes is 2 times the height of the function evaluated at 5, 7 and 9.
2 ·
( 1
2 ·^5 −^2
)
= 5 − 4 = 1
2 ·(^12 · 7 − 2 )= 7 − 4 = 3
2 ·(^12 · 9 − 2 )= 9 − 4 = 5
The approximate sum of the total area under the curve is:− 1 + 1 + 3 + 5 =8 square units.
Example B
Evaluate the exact area under the curve in Example A using the area formula for a triangle.
Solution:Remember that the area below thexaxis is negative while the area above thexaxis is positive.
Negative Area:^12 · 3 · 1. 5 =^94
Positive Area:^12 · 5 · 2. 5 =^254
Area under the curve between 1 and 8:^254 −^94 =^164 = 4