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### Course: Integral Calculus>Unit 1

Lesson 12: Definite integrals of common functions

# Definite integral of trig function

Sal finds the definite integral of 9sin(x) between 11π/2 and 6π.

## Want to join the conversation?

• at i don't understand why he can just subtract?
• That's a good question Vivian! See, he is not actually subtracting; he is putting it in terms that are easier to understand. Cosine is a periodic function with a period of 2π. This means that every 2π, the function repeats. This makes f(0) = f(2π) = f(4π) and etc. It is a ton easier to know the cosine of a more familiar value, namely those that are 0<x<2π. Sal simply rewrote 11π/2 as 3π/2 + 4π. Just like f(0) = f(2π), we can also say that f(3π/2) = f(7π/2) = f(11/2) and etc. I hope that helped!
• Is it acceptable to state that the antiderivative of sin(x) is -cos(x) instead of multiplying the sin(x) and 9 by -1? I believe it results in the same answer in this case.
• I can see no reason not to, as long as you keep careful track of the signs.
(I do it that way as well.)
• This feels wrong to me.

I understand that cos(6pi) = cos(0), and cos(11pi/2) == cos(3pi/2), but those are just the point values of the the functions. When we integrate, we're taking tiny slices and adding them together. Don't have to do this over the entire range 6pi to 11pi/2?

It seems like this shortcut is disregarding the fact that this is a sum.
• Because cos(x) is a constant oscillating function, the areas will keep on being the same because the area under a curve is - and above is + so because in intervals of pi/2, it is equal to each other if the intervals stay the same even if the bounds change
• So you can integrate in radians too?
• Actually it is easier to differentiate and integrate using radians instead of degrees. The formulas for derivatives and integrals of trig functions would become more complicated if degrees instead of radians are used (example: the antiderivative of cos(x) is sin(x) + C if radians are used, but is (180/pi)sin(x) + C if degrees are used). This is one of the main reasons why radian measurement is taught in trigonometry.
The simplicity of using radians instead of degrees should come as no surprise. Whereas the choice of 360 degrees in a circle is arbitrary, the fact that there are 2*pi radians (radiuses) in a circle is a consequence of the fact that the circumference is 2*pi*r. So radians, unlike degrees, is a natural unit of angle measurement.

Have a blessed, wonderful day!
• When do we know to use either radian or degree mode on our calculator when calculating integrals?
(1 vote)
• When calculating integrals, always use radians.
• In this example Sal evaluates from 11/2pi to 6pi. In earlier videos on integration with non trig functions we saw that evaluating an integral from, for example 10 to 2 (where 10 is at the buttom and 2 at the top) required us to swap the signs for the final result.

Does this not apply to integrals of periodic functions?
(1 vote)
• I think you only need to do that if you're using Riemann sums.
• Not sure why at he takes the answer to cos(3pi/2) and plugs that in for cos(11pi/2). I don't see how that works considering that he's ignoring the 4pi he subtracted to put the cosine in terms of 2pi. Wouldn't it be easier just to use a calculator instead of converting?
(1 vote)
• The 4pi can be ignored there, as the cosine of even multiples of pi is always 0. So, if you write 11pi/2 as (4pi + 3pi/2), you'll see that the 4pi throws you back on the positive x axis, and then the 3pi/2 moves you into the first quadrant, where cosine is positive. So, it boils down to just cos(3pi/2)

Yes, it would be easier to use a calculator. But, you won't have a calculator handy everytime. It's good to know how to break down big angles into smaller, calculable ones