Warmup 11/30/15 How was your Thanksgiving? What are you most thankful for in your life? To define the definite integralpp 251: 3, 4, 5, 6, 9, 11 Objective Tonight’s Homework

Homework Help Let’s spend the first 10 minutes of class going over any problems with which you need help.

The Fundamental Theorem of Calculus I We’re far enough along now to define the main, fundamental equation for calculus.

The Fundamental Theorem of Calculus I We’re far enough along now to define the main, fundamental equation for calculus. We’ve seen that we can calculate the area under a curve as a series of infinite rectangles.

The Fundamental Theorem of Calculus I We’re far enough along now to define the main, fundamental equation for calculus. We’ve seen that we can calculate the area under a curve as a series of infinite rectangles. Take a minute and do the following: Find the area under y = x from 0 to 5. Then find the integral of y = x. Plug in 5 for x after doing this and tell me what you get.

The Fundamental Theorem of Calculus I The Fundamental Theorem of Calculus If a function f(x) has an antiderivative ( F(x) ) over the range from a to b, then… f(x) dx = F(b) – F(a) ∫ baba

The Fundamental Theorem of Calculus I The Fundamental Theorem of Calculus If a function f(x) has an antiderivative ( F(x) ) over the range from a to b, then… f(x) dx = F(b) – F(a) We call the above operation a “definite integral” because we evaluate it with numbers to get an end result. Why is this so amazing? Because this value is also the area under the curve. ∫ baba

The Fundamental Theorem of Calculus I Example: Find the area under f(x) = sin(x) from 0 to π.

The Fundamental Theorem of Calculus I Example: Find the area under f(x) = sin(x) from 0 to π. f(x) dx sin(x) dx [-cos(x)] -cos( π ) - -cos(0) ∫ baba ∫ π0π0 π0π0

The Fundamental Theorem of Calculus I There’s another thing to note here. The definite integral can differentiate between area above the x-axis and area below the x-axis.

The Fundamental Theorem of Calculus I There’s another thing to note here. The definite integral can differentiate between area above the x-axis and area below the x-axis. Example: Calculate sin(x) ∫ 2π02π0

The Fundamental Theorem of Calculus I There’s another thing to note here. The definite integral can differentiate between area above the x-axis and area below the x-axis. Example: Calculate sin(x) [-cos(x)]-cos(2 π ) - - cos(0) ∫ 2π02π0 2π02π0

The Fundamental Theorem of Calculus I There’s another thing to note here. The definite integral can differentiate between area above the x-axis and area below the x-axis. Example: Calculate sin(x) [-cos(x)]-cos(2 π ) - - cos(0) How do we get 0 area?! ∫ 2π02π0 2π02π0

The Fundamental Theorem of Calculus I From 0 to π, our function is above zero, but from π to 2 π, our function spends just as much time below zero. These two portions cancel out. The first half is positive and the second half is negative.

The Fundamental Theorem of Calculus I One last idea for today. We defined this area under the curve initially as a sum of rectangles. But which rectangles? We used lower sums, but we could have used upper sums or even midpoint sums.

The Fundamental Theorem of Calculus I In the mid 1800s, a mathematician named Riemann realized that this was a problem. He wanted to see a generic way of getting sums.

The Fundamental Theorem of Calculus I In the mid 1800s, a mathematician named Riemann realized that this was a problem. He wanted to see a generic way of getting sums. With this in mind, he created the Riemann sum. This is a summing up of rectangles like we’ve seen before, but now, each rectangle can be any width and can touch the function at any point, even if every rectangle is different.

The Fundamental Theorem of Calculus I He summed this up with the following function: Riemann Sum = lim f(x i )Δx i Where || P || is the width of the widest partition. || P ||  0 ∑ i=n i=1

The Fundamental Theorem of Calculus I He summed this up with the following function: Riemann Sum = lim f(x i )Δx i Where || P || is the width of the widest partition. Example: Write e x dx as a Riemann sum || P ||  0 ∑ i=n i=1 ∫ 2020

The Fundamental Theorem of Calculus I He summed this up with the following function: Riemann Sum = lim f(x i )Δx i Where || P || is the width of the widest partition. Example: Write e x dx as a Riemann sum Riemann Sum = lim e x Δx i || P ||  0 ∑ i=n i=1 ∫ 2020 || P ||  0 i=n i=1 ∑ i

Group Practice Look at the example problems on pages 247 through 250. Make sure the examples make sense. Work through them with a friend. Then look at the homework tonight and see if there are any problems you think will be hard. Now is the time to ask a friend or the teacher for help! pp 251: 3, 4, 5, 6, 9, 11

Exit Question What will the result be if we integrate sin(x) or cos(x) from –x to x? a) 0 b) 1 c) π d) π/2 e) Not enough information f) None of the above