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## AP®︎/College Calculus BC

### Course: AP®︎/College Calculus BC>Unit 9

Lesson 9: Calculator-active practice

# Evaluating definite integral with calculator

Here we walk through how to use a graphing calculator to compute the integral found in the last video.

## Want to join the conversation?

• At what times can we actually use a calculator to evaluate integrals? They're (mostly) not allowed on tests...
• On the calculator section of the AP calc tests
• Can TI-84's do this? If so, I'd love the assistance!
• You can do this by pressing the 'math' button and scrolling down to fnint( ) (which is the 9th one down and will also come up by pressing '9'). This is the same function Sal uses on his 85 and once you select it the steps for inputting the function and other information are the same as in the video.
• Does this method work regardless of whether you are in Function or Polar? I use a TI-84+.
• At , Sal says that evaluating definite integrals with a calculator can be useful when you "can't actually evaluate them analytically". Are there integrals that are impossible to evaluate analytically? If so, can I see some examples?
(1 vote)
• ∫e^(x^2) dx cannot be expressed as an elementary function.
• why did you not multiply by 9/2 instead of just 9?
• He states in his previous video that the two areas he is finding are identical. If he was finding only one of those areas, he would multiply by 9/2 instead of 9, however he is looking for the area of the full region.
• Is it possible to cube root a number on khans' calculators?
• Does anyone know how to work this using a TI-83? I tried following what Sal did in the video, but I got an error message.
• Can a graphing calculator algebraically calculate an integral? So, essentially, just finding an antiderivative.
(1 vote)
• It can calculate a definite integral, but unfortunately it cannot find an indirect integral.
• When we try to find the limits of integration, why do we have to set r equal to zero.
Can someone explain this to me.
(1 vote)
• You set r = 0 when you cannot figure out the limits of integration just by inspection (just by looking at the graph). Usually that happens when at both bounds r = 0.
(1 vote)

## Video transcript

- [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.