Area defined by polar graphs
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Evaluating definite integral with calculator
- [Voiceover] In the last video, we tried to find the area of the region, I guess this combined area between the blue and this orange, the area, I guess the overlap between these two circles and we came up with nine pi minus 18 all of that over eight. What I want to do in this video, you could have also used a typical graphing calculator to come up with the same result, and it would have actually evaluated the definite integral. So let's see how you do that. Now this, what I'm doing here, you could do this for a traditional, what if you're dealing with Cartesian coordinates, rectangular coordinates, or for polar coordinates, cause it's really just about evaluating the definite integral. So we wanted to evaluate nine times the definite integral from zero to pi over four sin squared theta d theta, so how do I do that? Well I can go to second, calculus, then I do the F N INT, that's definite integral. So let's use that function, and then the first thing, you want to say "well, what are you taking "the definite integral of?" And we're taking the definite integral of... Sine, actually I want the parentheses, sine, and I could use any variable here, as long as I'm consistent with what I'm integrating with respect to. So I tend to use just the "x" button, because there is an "x" button, but we'll just assume that in this case x is theta. So sine of x squared, instead of sine of theta squared, we're once again assuming that x is equal to theta. And then the next one, you specify, "Well what's the variable you're taking "the integral with respect to?" In this case it's x, if we'd put in a theta here then we would want to put a theta there as well. And then you want the bounds of integration, and you should assume that your calculator, or if you're doing this, if you're in radiant mode, or if you're dealing with radiants, you should assume you're in radiant mode, I just did before I evaluated this. We're going between zero and pi over four. Zero and pi over four. And then, we get... So we get this number, and then we wanted to multiply it times nine. So my previous answer times nine. If I just press times, it does this, "Previous answer times" nine, is equal to this number, one point two eight four two nine. So let's verify that that's the same exact value we got when we actually evaluated the integral by hand. So if we take nine, nine pi minus 18 divided by eight, divided by eight, what do we get? We get the exact same value. So anyway, hopefully that's satisfying, that we got the same value either way, and a little exposure for how you might be able to evaluate some integrals using a calculator, which can be useful when you can't actually evaluate them analytically.