Working a Chapter 14 Problem

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Revised July 23, 2013
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The problem statement is:
Find the area of the region between `theta=0.5` and `theta=1.4`, and between `r=6+sin(12theta)` and `r=9+sin(18theta)`. Express your answer rounded to the nearest thousandths. [Two small reminders: `intsin^2(ax)dx\quad=\quad x/2-(sin(ax)cos(ax))/(2a)+C` and `intcos^2(ax)dx\quad=quadx/2+(sin(ax)cos(ax))/(2a)+C`]
The written solution to this is given in the pencast of this problem. That written solution ends with the task of evaluating `1/2(45theta - cos(18theta) + cos(12theta) - (sin(36theta)/(72) + sin(24theta)/48 |_0.5^1.4 )` To do this on the calculator we can assign 1.4 to x and then enter the entire expression, using `X` in place of `theta`. I did this, along with dividing the expression by 2 and storing the result in A. Here is an image of the calculator screen at that point.

Figure 1 Then, we can store the result in B for later use and follow that by storing the value 0.5 in X Here is an image of the calculator screen at that point.

Figure 2 Then we can recall the long expression (recall by repeating the use of the key sequence "2nd" "Enter") antd then execute that expression to get

Figure 3 At that point we just need to subtract that result from our earlier one, the values that we stored in B. Figure 4 shows that process:

Figure 4 This givew us the final answer.

As noted on the pencast, we did not need to do all of the work that we did. In fact we had no particular need to simplify some of the exressions. Furthermore, the TI-83/84 calulators will actually do the integration for us. Not the symbolic integration, but the it will compute the definite integral if we can get the problem in that form. In the pencast we made note of the point in the solution where we said that we needed to find `int_0.5^1.4 ((9+sin(18theta))^2)/2 - ((6+sin(12theta))^2)/2 d theta`. We can have the calculator do this for us. The first step is to find the right command. That command is fnInt( and it is on the MATH options page:

Figure 5 We select that command and then put in four arguments for the command

the expression of the function we are trying to integegrate, in this case that would be the expression for `((9+sin(18theta))^2)/2 - ((6+sin(12theta))^2)/2 `
the varible of integeration, in this case, `theta`
the numeric lower bound of integration, in this case 0.5
the numeric upper bound of integration, in this case 1.4

Figure 6 shows this with X substituted for `theta`.

Figure 6