# Unit 6: Integration and Accumulation of Change

### The Integral & Area Under A Curve

• Up to here we’ve learned about the derivative, the rate of change. Now we have the integral ∫ also called the antiderivative.

• The derivative shows us the change/unit

• So the antiderivative shows us the total change

• The first type is called a definite integral and shows us the area of the region under the function and the x-axis. It gives us the accumulation/total change!

• Let’s say we have a function that is shaped like that (or any function at all), if the definite integral needs us to get the area under the function, how would we do that?

• Because this is a shape that we have no formula for, we can estimate it using shapes that we do know

• We can split this area up into rectangles!

• The more rectangles we have, the better our estimate is!

• This method is called a Riemann sum!

### Riemann & Trapezoidal Sums

We can take a Riemann Sum from the left, or from the right!

• For left-handed sums we use the endpoints (number) on the left

• For right-handed sums we use the endpoints (number) on the right!

The formulas are the same for any rectangle, base * height!

• Take the width of your rectangle and multiply it by the height of the rectangle!

• Do this for each rectangle you have and add them all together

• To get these rectangles even more accurate, we can use a midpoint sum

• We still use the formula for a rectangle, but we use the value for the height in between!

• A shape that would more closely fit the shape of the curve is a trapezoid

• Therefore, we can use trapezoidal sums!

• We know that the formula for a trapezoid is (1/2)(b1 + b2)(h)

• For example, our second trapezoid would be (1/2)(2 + 5)(1)

• Still a width of 1 but we add the two heights!

Most of the time you are given a table to take a Riemann Sum from!

0

2

4

7

1

6

10

15

• Left Sum: (2)(1) + (2)(6) + (3)(10)

• Notice how this is a left sum so we don’t use the furthest right value

• Right Sum: (2)(6) + (2)(10) + (3)(15)

• Same thing for the right, except we don’t use the furthest left value

• Midpoint: (4)(6)

• Not complete but you see how we use a width of 4 and then the height in between

• Trapezoid: (1/2)(1+6)(2) + (1/2)(6+10)(2) + (1/2)(10+15)(3)

### Tabular Riemann Sums

• The majority of the time when you have to use a Riemann Sum, the AP gives it to you in tabular format

Years:(t)

2

3

5

7

10

Height:H(t)

1.5

2

6

11

15

• Trapezoids: (1/2)(1.5+2)(1) + (1/2)(2+6)(2) + (1/2)(6+11)(2) + (1/2)(11+15)(3)

• Left Sum: (1)(1.5) + (2)(2) + (2)(6) + (3)(11)

• Right Sum: (1)(2) + (2)(6) + (2)(11) + (3)(15)

• You do not have to simplify these!

### Fundamental Theorem of Calculus & Antiderivatives

• For differentials, you know that you had a set of different rules that you can use to take the derivative. The same applies for the antiderivative!

• Intuitively, it’s the opposite of what you do to take a derivative EXCEPT…

• We can only use the power rule!

• If the power rule for a derivative tells us to multiply down and decrease the power, then the opposite of that would be to divide and increase the power!

• The +C is very important!

• If you take the derivative of any number without an x, you get zero

• Therefore if we’re doing the reverse process, we don’t know what this number could be, therefore we add on a C for the constant of integration

• The integral of 2x is really 2x^2/2 but that simplifies to x^2!

• Remember that if the integral is not in power rule format, we must algebraically manipulate it so that we can use the power rule

• The two numbers at the top and bottom of the integral means that it is a definite or bounded integral

• It means we are trying to find the area below 2 and 3

• Because we have a function, we don’t have to graph it out, instead we have something called the First Fundamental Theorem of Calculus

• (8 chapters in and you’re just now learning about the thing fundamental to calculus huh)

The first fundamental theorem says that the integral from a to b is equal to the antiderivative, plug in b, and then plug in a and subtract!

• Let’s say that we had ∫2x from before (from x=2 to x=3), according to the first fundamental theorem it is equal to (3)^2 - (2)^2

• Sometimes, getting an integral into power rule format is nearly impossible, in those cases there are other techniques we can do!

• If your integral contains trigonometry, the best thing to do is just memorize the derivative of trig functions, and the integral will be the opposite

• Ex. d/dx sinx = cosx

• Therefore, ∫cosx = sinx

• You can manually derive these but because this is a timed AP exam it’s more efficient to memorize these!

• Your other option is U-substitution!

1. Chose a term to be your “u”

2. Take the derivative of this value to get du/dx

3. Substitute in your u value for the term and your du/dx value for dx

4. Take the integral

• U-substitution is tricky but helpful for some problems!

• You got this!!! 👍

1. Ex. ∫(x - 4)^10

2. Let u = x-4

3. du/dx = 1

4. dx = du/1

5. ∫(u)^10 du

6. u^11/11 + C

7. (x-4)^11/11 + C