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5.1: Exploring Area under the Curve

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This activity is intended to supplement Calculus, Chapter 4, Lesson 3.

Problem 1 – Explore and discover

Graph the curve y = x^2.

Your challenge is to think of at least two ways to estimate the area bounded by the curve y = x^2 and the x-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 x-axis

Graph y = x^2 and set your window to [-0.1, 1] for x and [-0.2, 1.3] for y. 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 y = x^2.

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 x-value and the height is the y-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:

R_5 &= 0 . 2 \cdot f1(0 . 2) + 0 . 2 \cdot f1(0 . 4) + 0 . 2 \cdot f1(0 . 6) + 0 . 2 \cdot f1(0 . 8) + 0 . 2 \cdot f1(1 . 0)\\& \qquad \qquad \qquad \qquad \qquad \qquad \text{or}\\R_5 &= 0 . 2 [f1(0 . 2) + f1(0 . 4) +  f1(0 . 6) +  f1(0 . 8) +  f1(1 . 0)]

  • 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 \sum_{x=1}^5x^2 form. Adjust what is being summed.

To sum it on the calculator, use Home > F3:Calc > 4:Sigma for the command with the format:

\sum(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 y = x^2.

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 x-value and the height is the y-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:

L_5 &= 0 . 2 \cdot f1(0) + 0 . 2 \cdot f1(0 . 2) + 0 . 2 \cdot f1(0 . 4) + 0 . 2 \cdot f1(0 . 6) + 0 . 2 \cdot f1(0 . 8)\\& \qquad \qquad \qquad \qquad \qquad \text{or}\\L_5 &= 0 . 2 [f1(0) + f1(0 . 2) + f1(0 . 4) + f1(0 . 6) + f1(0 . 8)]

  • 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 y = x^2?

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 x-value and the height is the y-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:

M_5 &= 0 . 2 \cdot f1(0 . 1) + 0 . 2 \cdot f1(0 . 3) + 0 . 2 \cdot f1(0 . 5) + 0 . 2 \cdot f1(0 . 7) +  0 . 2 \cdot f1(0 . 9)\\ & \qquad \qquad \qquad \qquad \qquad \text{or}\\M_5 &= 0 . 2 [f1(0 . 1) + f1(0 . 3) + f1(0 . 5) + f1(0 . 7) + f1(0 . 9)]

  • 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 y = x^2 on the interval [0, 1] is  \frac{1}{3} 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 \frac{1}{3}? Test your conjecture by using evaluating a sum that produces a much more accurate answer.

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Date Created:

Feb 23, 2012

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Aug 19, 2014
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