Unit 6: Integration and Accumulation of Change (copy)

The Integral & Area Under A Curve

• The integral ∫ is also called the antiderivative.

• The derivative shows us the change/unit, so the antiderivative shows us the total change.

• A definite integral shows us the area of the region under the function and the x-axis.

• We can estimate the area using shapes that we do know by splitting this area up into rectangles, using the Riemann sum method. The more rectangles we have, the better our estimate is!

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!

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

• The formula for a trapezoid is (1/2)(b1 + b2)(h)

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

• 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

• The antiderivative is the opposite of derivative!

• 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! The constant of integration.

• 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!

Advanced Integration

• If your integral contains trigonometry, memorize the derivative of trig functions, and the integral will be the opposite

• Ex. d/dx sinx = cosx, Therefore, ∫cosx = sinx

• 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

• Ex. ∫(x - 4)^10

  • Let u = x-4

  • du/dx = 1

  • dx = du/1
    *