Accumulated Change vs. Definite Integral
Martyna Wiacek MTH 116 C- Applied Calculus 11/6/2012 Chapter 5 Writing Assignment There is a correlation between area, accumulated change, and the definite integral that we have focused on throughout Chapter 5 in Applied Calculus. When looking at one rate-of-change function, the accumulated change over an interval and the definite integral are equivalent, their values could be positive, negative or zero. However, the area could never be negative because area is always positive by definition. The accumulated change looks at the whole area of the function that is between the graph and the horizontal axis.
For instance, if f (x) is a rate-of-change function the area between f (x) and the x-axis represents the accumulated change between x = a and x = b. However, the definite integral puts specific limits into the function and the area of a particular region can be determined. For example, if f (x) is a rate-of-change function it means that: is what you can consider the area. The accumulation of change in a certain function can be evaluated by using the area of the region between the rate-of-change curve and the horizontal axis.
We also see a similar relationship between the rate-of-change graph and the accumulated graph that we saw in derivatives. A minimum in the accumulated graph is caused by the rate-of-change function crossing over from positive to negative. A maximum in the accumulated graph is a result of the rate-of-change function moving from negative to positive. When there is a maximum or minimum in the rate-of-change graph you get an inflection point in the accumulation graph as well. Also, we see that if the rate-of-change function is negative then the accumulated graph is negative and so the accumulation graph is decreasing.
However, when the rate-of-change graph is increasing, it does not affect whether or not the accumulated graph is increasing or decreasing. There are several problems in our book that demonstrate this relationship. A specific example that I believe did a good job demonstrating it was: The graph in the figure represents the rate of change of rainfall in Florida during a severe thunderstorm t hours after the rain began falling: Part A: Use a grid to count boxes and estimate the accumulated area from 1 to x for values of x spaced 1 hour apart, starting at 0 and ending at 6.
Record the estimates in a table. 0| 0| 1| . 4| 2| . 65| 3| 1| 4| 1. 35| 5| 2| 6| 2. 4| Part B: Sketch the graph of the accumulation function based on the table values: Part C: Write the mathematical notation for the function sketched in part b: Part D: Write a sentence of interpretation for the accumulation form 0 to 6 hours: After 6 hours of rainfall in Florida, the amount of rain should accumulate to an estimate of 2. 4 inches.