Estimating average value with trapezoidal sum AP Calculus AB free response 2011 2b

0:00- [Instructor] And then part B, using correct units, explain the meaning of one over 10, one over 10 times the definite integral from zero to 10 of H(t)dt in the context of this problem. Use a trapezoidal sum with the four subintervals indicated by the table to estimate this thing.

0:17So, the integral from zero to 10 of H(t) is really the area of this entire, under this curve right over here. That's the integral, that's this part right over here. And then we're going to divide that, we're going to divide that by, we're going to divide that by 10.

0:33So, what this is really giving us, this expression right over here is the average, this is the average temperature. This is the average temperature over that time period, the integral of the temperature function divided by the total amount of time that has elapsed.

0:52And then using correct units, well, it's the average temperature, the average temperature is once again going to be in degrees Celsius. And it makes sense 'cause this right over here is in degrees Celsius. You're multiplying it by time right over here, but then you could divide by, you're dividing, I'm assuming by time right over here.

1:09So, this is times minutes divided by minutes. So, then you just get degrees Celsius again. So, this expression right over here is just the average temperature over those 10 minutes. Then they say, use a trapezoidal sum with the four subintervals indicated by the table to estimate this.

1:26So, the trapezoidal sum is we don't know the exact function here, so we won't be able to analytically evaluate this definite integral, but what we can do is divide this area into four sections. They tell us to use this, four subintervals, and we'll essentially divide it into four trapezoids.

1:43So, this is one trapezoid right over here. So, this is one trapezoid right over there. That's my first trapezoid from zero to two on the base. And you see, the left end of that trapezoid its height is 66, the right end is 60. And then the next trapezoid will go from two to five.

2:02Let me do that in a more... In a color that contrasts a little bit better. The next trapezoid will go from two to five. Next trapezoid goes from two to five. And then the third trapezoid goes from five to nine. The third trapezoid goes from five to nine. And then the fourth trapezoid goes from nine to 10, goes from nine to 10.

2:24And so, if I want to approximate the definite integral part right here, before we divide by 10, I just need to find the area of these four trapezoids. So, let's do that. The area of this first trapezoid is going to be the base, which is two times the average height.

2:42The height on the left side of this trapezoid is 66. The height on the right side of this trapezoid is 60. The average height, right over here, is going to be 63, just the average of 60 and 66. So, this area right over here is, this is 126. That's the area of the green part right over there.

3:02By the same logic, the area of this orange part, I want to do that in the orange color. The area of this orange part, the base right over here is three, base is three. And then the average height, the height over here is 60. The height over here is 52. This is, let's see, they're either eight apart.

3:20So, the average is going to be four away from each of it. It's going to be 56. So this is, the area here, is going to be three times three times 56, which is 150 plus 18. 150 plus 18 is 168. That's the area of this orange trapezoid. And then the area of this blue trapezoid, the base right over here is four.

3:45And then the average height, the height here is 52, the height here is 44. The average of 52 and 44, let's see, they're 10, they are, sorry, they are eight apart, eight apart. So then, it's just going to be four from each of these. So, it's 48. So, the average height here is 48.

4:01And so, four times 48 is 160 plus 32. So, it's 192. And then finally this last trapezoid, its base is only one. Its base, right over here, is only one. And then its height is 43 at the left side, 43 at the right side. So, its average height is 43.5, 43.5. So, it's 43.5 times one.

4:28So, that's just going to be 43.5. And so, if we add up the areas under or the areas of all these trapezoids, we have a pretty good approximation for the definite integral. I'll just use the calculator for this part right over here. So, we have 126 plus 168 plus 192 plus 43.5, 43.5 gives us 529.5.

4:58So, this is equal to 529.5. And so, that's our approximation for the definite integral part, but then we also have to divide it by 10 or we have to multiply it times one 10th. So, we've evaluated, so this part right over here, we got 529.5 as our approximation.

5:16It's not going to be exact, but using the trapezoidal sum, the sum of the areas of these trapezoids. And now we have to multiply it by one 10th. So, let's do that. So, or we can just divide by 10, which actually we don't need a calculator for that. 52.95. So, this whole thing evaluates to 52.95.