Calculating areas between curves with calculator

0:00- [Instructor] We're told let f of x equal all of this, g of x equal all of this. Let R and S be the two regions enclosed by the graphs of f and g as shown in the graph. All right, here, g is above f, here, f is above g for region S. Find the sum of the areas of regions R and S.

0:17Use a graphing calculator and round your answer to three decimal places. Well, thank goodness we're using a graphing calculator 'cause this could get quite hairy if we didn't. I have a graphing calculator here. Let's turn it on. And first, lemme make sure I'm in radian mode.

0:30On the AP calculus exam, you want to be in radian mode. Lemme click mode. And I am indeed in radian mode. And now I'm gonna take a definite integral and I'm just gonna go all the way from zero to four, x equals zero, all the way to four. I'm integrating with respect to x.

0:47If I was doing it by hand, I would have to figure out what this point of intersection is, or even if you're using a calculator. But with the calculator, what you could do is just take the absolute value of the difference of these functions. So regardless of which one is greater than the other one at a given x, you're always gonna get positive values.

1:03And so then that allows you to integrate all the way from zero to four and these things won't cancel each other out a bit. So let's now take our definite integral. So math, and before the AP exam, you should know your calculator really, really, really well. But that's a definite integral right over there, gives us this nice little integral looking thing.

1:23We're gonna go from x equals zero to x equals four, zero to four. And then as I mentioned, I'm gonna take the absolute value of the difference between these functions. So let me go again to math, then number, absolute value is the first choice there. Enter, and let's see, I'll just do f of x minus g of x.

1:47So we're gonna have two times cosine of pi, second pi, times the square root of x. And now I can close that parentheses. Yep, two times cosine times pi times the square root of x, close parentheses. And then from that I am going to subtract, minus g of x. Oh, this is painful. All right.

2:18Three divided by eight times x squared minus three divided by two times x. Oh, let me be careful here because I'm subtracting this. So if I'm subtracting a positive term here, subtracting this negative will become a positive. Oh, I almost made that mistake. So this one right over here, this will be plus 3/2, plus 3/2, times, lemme see what's going on here.

2:53Oh, there, that's where I am. 3/2 times x, 3/2 times x, plus two, plus two. And let's see, and then there should be just a closed parentheses, yep, that's the absolute value sign. All right. And everything is going to be with respect to x. So let me just check everything I did before I press enter.

3:16So my integral goes from zero to four of the absolute value of two cosine pi times the square root of x, and then minus 3/8 x, minus three divided by eight times x squared, plus 3/2 times x. Oh, I almost made another mistake. I'm subtracting this whole thing. So this has to be minus two at the end.

3:41Good thing I double checked my spider sense, served me well, dx. All right, I think I got it now. Click enter. It's chugging for a little bit. It amazes me that this technology really hasn't changed much since I was in school. That it's pretty much the same calculator.

3:59If we're, and if we wanna round our answer to three decimal places, it would be 5.039. And we would be done.