14.9. Area Under a Curve http://www.ck12.org
The blue approximation uses right handed boxes. The red approximation assigns the height of the box to be the
minimum value of the function in each subinterval. The green approximation assigns the height of the box to be the
maximum value of the function in each subinterval. The yellow approximation uses left handed boxes. Rectangles
above thex-axis will have positive area and rectangles below thex-axis will have negative area in this context.
All four of these area approximations get better as the number of boxes increase. In fact, the limit of each approxi-
mation as the number of boxes increases to infinity is the precise area under the curve.
This is where the calculus idea of an integral comes in. An integral is the limit of a sum as the number of summands
increases to infinity.
∫f(x) =lim
n→∞
n
i∑= 1 (Area o f box i)
The symbol on the left is the calculus symbol of an integral. Using boxes to estimate the area under a curve is called
a Riemann Sum.
Example A
Use four right handed boxes to approximate the area between 1 and 9 of the functionf(x) =^12 x−2.
Solution: