6.1: The Area Between Curves
This activity is intended to supplement Calculus, Chapter 5, Lesson 1.
In this activity, you will explore:
- Using integrals to find the area between two curves.
Use this document to record your answers.
Problem 1 – Making Sidewalks
While integrals can be used to find the area under a curve, they can also be used to find the area between curves through subtraction (just make sure the subtraction order is the top curve minus bottom curve.)
Suppose you are a building contractor and need to know how much concrete to order to create a pathway that is \begin{align*}\frac{1}{3}\end{align*}
\begin{align*}f(x) = \sin(0.5x) + 3\end{align*}
Graph both functions. Adjust the window settings to \begin{align*}-6.4\le x \le 6.4\end{align*}
Now use the Integral tool from the Math menu to calculate the integrals of \begin{align*}f(x)\end{align*}
Enter \begin{align*}-2\pi\end{align*}
- What is the value of the integral of \begin{align*}f(x)\end{align*}
f(x) ? Of \begin{align*}g(x)\end{align*}g(x) ?
On the Home screen, define \begin{align*}f(x)\end{align*}
Hint: the area is equal to the integral of \begin{align*}f(x) - g(x)\end{align*}
Note: The nInt command has the syntax: nInt(function, variable, left limit, right limit)
- What is the formula for the volume of the sidewalk?
- Now calculate how much concrete is needed for the pathway.
Problem 2 – Finding New Pathways
The owners have changed the design of the pathway. It will now be modeled from -2 to 2 by:
\begin{align*}f(x) & = x(x + 2.5)(x - 1.5) + 3 \\
g(x) & = x(x + 2)(x - 2) \end{align*}
Graph both functions. Adjust the window settings to \begin{align*}-4 \le x \le 4\end{align*}
Calculate the integrals of \begin{align*}f(x)\end{align*} and \begin{align*}g(x)\end{align*}. Enter \begin{align*}-2\end{align*} for the lower limit and \begin{align*}2\end{align*} for the upper limit.
- What is the value of the integral of \begin{align*}f(x)\end{align*}? Of \begin{align*}g(x)\end{align*}?
- Now calculate how much concrete is needed for the pathway on the Home screen. Remember to define \begin{align*}f(x)\end{align*} and \begin{align*}g(x)\end{align*}.
Problem 3 – Stepping Stones
The owners also want stepping stones, which can be modeled by
\begin{align*}f(x) & = -(x - 1)(x - 2) + 2 \\ g(x) & = (x - 1)(x - 2) + 0.5. \end{align*}
This situation different because the starting and stopping points are not given. Assume that the stepping stones are \begin{align*}\frac{1}{3}\end{align*} foot thick.
Graph both functions. Adjust the window settings to \begin{align*}-1\le x \le 7\end{align*} and \begin{align*}-4 \le y \le 4\end{align*} with a step size of \begin{align*}0.5\end{align*} for both. Use the Intersection tool in the Math menu to find the intersection points. You can also use the Solve command on the Home screen.
- What are the coordinates for the two intersection points?
Calculate the integrals of \begin{align*}f(x)\end{align*} and \begin{align*}g(x)\end{align*}. Use the \begin{align*}x-\end{align*}values of the intersection points as the lower and upper limits.
- What is the value of the integral of \begin{align*}f(x)\end{align*}? \begin{align*}g(x)\end{align*}?
- Now calculate how much concrete is needed for the pathway on the Home screen. Remember to define \begin{align*}f(x)\end{align*} and \begin{align*}g(x)\end{align*}.
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