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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

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!

  • 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!!! 👍

    • Ex. ∫(x - 4)^10

      • Let u = x-4

      • du/dx = 1

      • dx = du/1

      • ∫(u)^10 du

      • u^11/11 + C

      • (x-4)^11/11 + C

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

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!

  • 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!!! 👍

    • Ex. ∫(x - 4)^10

      • Let u = x-4

      • du/dx = 1

      • dx = du/1

      • ∫(u)^10 du

      • u^11/11 + C

      • (x-4)^11/11 + C

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