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

Advanced Integration

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!