5.1: Exploring Area under the Curve
This activity is intended to supplement Calculus, Chapter 4, Lesson 3.
Problem 1 – Explore and discover
Graph the curve .
Your challenge is to think of at least two ways to estimate the area bounded by the curve and the axis on the interval [0, 1] using rectangles. Use the following guidlines:
- all rectangles must have the same width
- you must build all your rectangles using the same methods
- the base of each rectangle must lie on the axis
Graph and set your window to [-0.1, 1] for and [-0.2, 1.3] for . Draw your first and second method on the graphs below. For each method calculate the following:
- Number of rectangles
- Height and width of each one
- Area of each
- Sum of the area
- Which method did a better job?
- How could you improve on it?
In the following problem, you will examine three common techniques that use rectangles to find the approximate area under a curve. Perhaps you discovered some of these techniques during your exploration in the above problem. The first problem uses rectangles whose right-endpoints lie on the curve .
Problem 2 – Using five right-endpoint rectangles
Divide the interval [0, 1] into five equal pieces. Enter the information for each interval or rectangle in the table below. Remember that the right endpoint is the value and the height is the value of the right endpoint on the curve.
Interval | Right Endpoint | Height | Area |
---|
- Calculate this sum.
- Now add up the numbers in the Area column.
The formula that can be used to express the total area is:
- Are these two numbers the same or different?
Another way to find the area of the rectangles is using sigma notation.
- Write the notation in the form. Adjust what is being summed.
To sum it on the calculator, use Home > F3:Calc > 4:Sigma for the command with the format:
(expression, variable, lower limit, upper limit)
- Does this agree with the answer for the area you found previously?
Problem 3 – Using five left-endpoint rectangles
This problem uses rectangles whose left-endpoints lie on the curve .
Divide the interval [0, 1] into five equal pieces. Enter the information for each interval in the table below. Remember that the left endpoint is the value and the height is the value of the left endpoint on the curve.
Interval | Left Endpoint | Height | Area |
---|
- Calculate this sum.
- Now add up the numbers in the Area column.
The formula that can be used to express this area is:
- Are these two numbers the same or different?
- What is the sigma notation for the area of the rectangles?
- Use the calculator to find the sum. Does this result agree with the answer above?
Problem 4 – Using five midpoint rectangles
We will now investigate a midpoint approximation. How would you draw five rectangles, with equal width, such that their midpoints lie on the curve ?
Divide the interval [0, 1] into five equal pieces. Enter the information for each interval or rectangle in the table below. Remember that the midpoint is the value and the height is the value of the midpoint on the curve.
Interval | Midpoint | Height | Area |
---|
- Calculate this sum.
- Now add up the numbers in the Area column.
- Are these two numbers the same or different?
The formula that can be used to express this area is:
- What is the sigma notation for the area of the rectangles?
- Use the calculator to find the sum. Does this result agree with the answer above?
Problem 5- Summarize your findings
In this activity, you explored three different methods for approximating the area under a curve. The exact area under the curve on the interval [0, 1] is or 0.333.
- Which approximation produced the best estimate for the actual area under the curve?
- Describe which factors contribute to left, right, and midpoint rectangles giving overestimates versus underestimates.
- What can you do to ensure that all three of these techniques produce an answer that is very close to ? Test your conjecture by using evaluating a sum that produces a much more accurate answer.