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!

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  • 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!

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  • The more rectangles we have, the better our estimate is!   * This method is called a Riemann sum!

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Riemann & Trapezoidal Sums

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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

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  • 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!

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  • 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!

0247
161015
  • 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)

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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)235710
Height:H(t)1.5261115
  • 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

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  • 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)

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 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

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

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  • 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!

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  • Your other option is U-substitution!

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  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

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