Quarter Three Benchmark

Approximating/Finding Areas Under Curves

By Nathan Kamal and Rick Kinard

Hello! Welcome to our calculus benchmark. This qfjgoogroduction to integrals, and methods to approximate the area under a curve. The assignment was to select a set of data from any real-world situation with an independent and dependent variable, the dependent variable being a rate. We chose acceleration, the relationship between velocity and time, using a set of data collected from a 2010 Artega GT. Below is a v/t chart and graph.

V mph |
30 |
40 |
50 |
60 |
70 |
80 |
90 |
100 |
110 |
120 |
130 |
140 |
150 |
160 |
170 |

T sec |
1.3 |
1.7 |
2 |
2.4 |
3 |
3.5 |
4 |
4.8 |
5.5 |
6.3 |
7.2 |
8.3 |
11 |
14 |
18 |

Now that we can visualize our data in the form of a graph, we can find the area under the curve over a given interval to calculate distance. Because a distance/time function is the derivative function of velocity/time function, we can use integrals to find how far the car has travelled, by using a t/v graph (in this case, the graph is actually TIME as a function of VELOCITY, due to the fact that time is a fixed, independent variable). We can approximate the area by using trapezoids and rectangles.