Integrals NO CALCULATOR TEST Chapter 5

Riemann Sums 5.1

5.1 Riemann Sums

 Left-hand Riemann Sum: Finding the area under the curve using rectangles the same height as the left end of the interval.  Let’s pretend n = 5 in the last example  Right-hand Riemann Sum: Finding the area under the curve using rectangles the same height as the right end of the interval.  Let’s pretend n = 5 again  Midpoint Riemann Sum: Finding the area under the curve using rectangles the same height as the midpoint of the interval.  Let’s continue pretending n = 5  CTM – Without Riemann?

5.1 Riemann Sums

 A note on estimates:  If a function is monotonic and increasing, the left-hand Riemann Sum is an underestimate and the right-hand Riemann Sum is an overestimate.  If a function is monotonic and decreasing, the left-hand Riemann Sum is an overestimate and the right-hand Riemann Sum is an underestimate.  If the function is not monotonic, you cannot say if you are under- or over-approximating.  Assignment: pg. 270 (1-6, 15-19, 26-36)

An Introduction to Definite Integrals (and anti-derivatives) 5.2 (and 4.2)

5.2 Intro to Definite Integrals  As we saw in 5.1, using a larger number of rectangles to estimate area under the curve increases our accuracy by having smaller spaces above or below the curve. What if we wanted to find the exact area?  In order to have the exact area, we need the number of rectangles to approach infinity.

5.2 Intro to Definite Integrals

From 4.2 Anti-derivatives

From 4.2 Anti-Derivatives

Properties of Integrals and the Fundamental Theorem of Calculus 5.3 (part 1)

5.3 Properties of Integrals

5.3 Fundamental Theorem of Calculus

Mean Value Theorem (revisited) 5.3 (part 2)

5.3 Mean Value Theorem

5.3 Mean Value Theorem Project  1. Choose a “curvy” fourth degree polynomial that is not symmetric about a point.  2. Graph your polynomial as accurately as possible on graph paper for an “interesting” interval of at least 10 units long (for part 3A, it might be easier with two or more copies of this graph). [Graphsketch.com?]  [Hopefully, we will be using the Chromebooks in class or going to a computer lab one day to work on this.]

5.3 Mean Value Theorem Project  3. Apply the MVT to your graph on your interval two ways:  A)  Rausch’s Geometric Method: By constructing the (b-a) x f(c) rectangle and “filling in” the empty sections in your rectangle with the parts above it. Explain why this is a consequence of the MVT.  or  Gentil’s Geometric Method: By constructing the (b-a) x f(c) rectangle and filling in the rectangle and the area under the curve with the same small items and comparing the number of items.

5.3 Mean Value Theorem Project  AND  B) Fraunhoffer & Hubbert’s Analytic Method: By using integrals instead of cut and paste considering the function f(x) – f(c). Your focus should be on explaining WHY you are calculating everything you are calculating as much as your actual calculations.  or  Chen’s Analytic Method: By dividing up the original (b-a) x f(c) rectangle and finding the area in chunks between values of c.  4. Correctly label all of the geometry from the MVT on your picture(s). [a, b, b-a, f(c), c, f(x),, (b-a)f(c), etc.]

5.3 Mean Value Theorem

Second Fundamental Theorem of Calculus (plus more FTC) 5.4

Second Fundamental Theorem

Fundamental Theorem of Calc

Trapezoidal Rule and Simpson’s Rule 5.5

5.5 Trapezoidal Rule  The Trapezoidal Rule estimates the area under the curve using n trapezoids instead of n rectangles, increasing the accuracy of the estimate.  NOTE: Using the Trapezoidal Rule on a graph that is always concave up on the interval will give an over-estimate.  NOTE: Using the Trapezoidal Rule on a graph that is always concave down on the interval will always give an under- estimate.

5.5 Trapezoidal Rule & Simpson’s Rule

5.5 Trapezoidal Rule (2011 AB #2b)

5.5 Trapezoidal Rule & Simpson’s Rule