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 \begin{align*}y = x^2\end{align*}.
Your challenge is to think of at least two ways to estimate the area bounded by the curve \begin{align*}y = x^2 \end{align*} and the \begin{align*}x-\end{align*}axis on the interval [0, 1] using rectangles. Use the following guidelines:
- 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 \begin{align*}x-\end{align*}axis
Graph \begin{align*}y = x^2\end{align*} and set your window to [-0.1, 1] for \begin{align*}x\end{align*} and [-0.2, 1.3] for \begin{align*}y\end{align*}. 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 \begin{align*}y = x^2\end{align*}.
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 \begin{align*}x-\end{align*}value and the height is the \begin{align*}y-\end{align*}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:
\begin{align*}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)]\end{align*}
- 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 \begin{align*}\sum_{x=1}^5x^2\end{align*} form. Adjust what is being summed.
To sum it on the calculator, use Home > F3:Calc > 4:Sigma for the command with the format:
\begin{align*}\sum\end{align*}(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 \begin{align*}y = x^2\end{align*}.
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 \begin{align*}x-\end{align*}value and the height is the \begin{align*}y-\end{align*}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:
\begin{align*}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)]\end{align*}
- 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 \begin{align*}y = x^2\end{align*}?
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 \begin{align*}x-\end{align*}value and the height is the \begin{align*}y-\end{align*}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:
\begin{align*}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)] \end{align*}
- 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 \begin{align*}y = x^2\end{align*} on the interval [0, 1] is \begin{align*} \frac{1}{3}\end{align*} 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 \begin{align*}\frac{1}{3}\end{align*}? Test your conjecture by using evaluating a sum that produces a much more accurate answer.
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