Definite integral of radical function | AP Calculus AB | Khan Academy

So we want to evaluate the definite integral from -1 to 8 of 12 * the cube root of x dx. Let's see, this is going to be the same thing as the definite integral from -1 to 8 of 12 * the cube root is the same thing as saying x to the 1/3 power dx.

And so now if we want to take the anti-derivative of this stuff on the inside, we're just going to do essentially the power rule. You could use the power rule of integrals, or it's the reverse of the power rule for derivatives, where we increase this exponent by one and then we divide by that increased exponent.

So this is going to be equal to 12 * x to the 1/3 + 1. Let me let me do that in another color just so we can keep track of it: x to the 1/3 + 1. And then we're going to divide by 1/3 + 1.

And so what's 1/3 + 1? Well, that's 4/3. 1/3 + 3/3, that's 4/3. So I could write it this way: I could write this as x to the 4/3 divided by 4/3.

And this is going to be, and I'm going to evaluate this at the bounds, so I'm going to evaluate this at, and I'll do this in different colors, I'm going to evaluate it at 8 and I'm going to evaluate it at -1. And I'm going, I'm going to subtract it evaluated at -1 from this expression evaluated at 8.

And so what is this going to be equal to? Well actually, let me simplify a little bit more. What is 12 divided by 4/3? So 12, I'll do it right over here. 12 over 4/3 is equal to 12 * 3/4, which we could use 12 over 1 * 3/4. 12 / 4 is 3, so this is going to be equal to 9.

3/4 of 12 is 9, so we could rewrite this. We could write this as 9 * x to the 4/3 power. So if we evaluate it at 8, this is going to be 9 * 8 to the 4/3 power. And from that, we're going to subtract it evaluated at -1. So this is going to be 9 * (-1) to the 4/3 power.

So what is 8 to the 4/3 power? I'll do it over here. So 8 to the 4/3 is equal to (8 to the 1/3) to the 4th power. These are just exponent properties here. 8 to the 1/3, the cube root of 8 or 8 to the 1/3 power, that's 2 because 2 to the 3 power is 8.

And 2 to the 4th power, well 2 to the 4th power is equal to 16, so 8 to 4/3 is 16. And what's (-1) to the 4/3? We'll say (-1) to the 4/3 is equal to -1. There are several ways you could do it; you can say -1 to the 4th and then the cube root of that, or the cube root of negative 1 and then raise that to the 4th power either way.

So let's do it the first way: 1 to the 4th and then take the cube root of that. Well negative 1 to the 4th is just 1, and then 1 to the 1/3 power, well that's just going to be equal to 1. So what we have here in blue, that's just equal to 1.

So we have 9 * 16 - 9 * 1. Well, that's just going to be 9 * 15. We have 16 - 9, and then we're going to take away a 9, so that's going to be 9 * 15.

So what is that? That is going to be equal to 9 * 15, which is 90 + 45, which is equal to 135. 135, and we're done.