Calculus I (MAT 145) Dr. Day Wednesday April 17, 2019 Chapter 5: Integrals and Anti-Derivatives Recovering a Function Knowing its Derivative Riemann Sums Riemann Sums with an Infinite Number of Subdivisions Definite Integrals and Indefinite Integrals: Connecting Derivatives and Anti- Derivatives The Fundamental Theorem of Calculus Part 1 Part 2 Monday, April 15, 2019 MAT 145

Antiderivatives, Integrals, and Initial-Value Problems Knowing f ’, can we determine f ? General and specific solutions: The antiderivative. The integral symbol: Representing antiderivatives Initial-Value Problems: Transforming a general antiderivative into a specific function that satisfies the given information. Read this: “The antiderivative of 2x with respect to the variable x” Wednesday, April 17, 2019 MAT 145

Accumulate, Accumulate, Accumulate! How much snow fell? Wednesday, April 17, 2019 MAT 145

Accumulate, Accumulate, Accumulate! How much snow fell? Wednesday, April 17, 2019 MAT 145

Accumulate, Accumulate, Accumulate! How much snow fell? Wednesday, April 17, 2019 MAT 145

Accumulate, Accumulate, Accumulate! How much snow fell? Wednesday, April 17, 2019 MAT 145

Accumulate, Accumulate, Accumulate! How much snow fell? Wednesday, April 17, 2019 MAT 145

Accumulate, Accumulate, Accumulate! How much snow fell? Wednesday, April 17, 2019 MAT 145

Areas and Distances (5.1) Use What You Know to Get at What You’re Looking For Choosing Endpoints Notation Accumulations From Rates Wednesday, April 17, 2019 MAT 145

Approximating Area: Riemann Sums To generate a way to calculate the area under the curve of a rate function, in order to determine an accumulation, we begin with AREA APPROXIMATIONS. We create something called a Riemann Sum and use better and better area approximations that will lead to exact area. Wednesday, April 17, 2019 MAT 145

Approximating Area: Riemann Sums Riemann Sum Applet Wednesday, April 17, 2019 MAT 145

Wednesday, April 17, 2019 MAT 145

Riemann Sums Calculate the exact value of a Riemann Sum to approximate the area under the curve y = x2-4 for 1 ≤ x ≤ 4, using n = 3 rectangles and using midpoints. Show a graph of the function that includes a sketch of your rectangles. Show all calculations using exact values. Clearly indicate the value of your Riemann Sum. Wednesday, April 17, 2019 MAT 145

Riemann Sums Suppose y = f(x) = x2-4 is an object’s velocity function. On 1 ≤ x ≤ 4: What is the NET CHANGE IN POSITION? What is the TOTAL DISTANCE TRAVELED? If 𝐹 𝑡 = 1 𝑡 𝑓 𝑥 𝑑𝑥 , What is 𝐹′ 𝑡 ? Wednesday, April 17, 2019 MAT 145

upper limit of integration differential This is called a definite integral. It includes lower bounds and upper bounds that represent boundary values of an x-axis interval. integrand lower limit of integration Wednesday, April 17, 2019 MAT 145

Wednesday, April 17, 2019 MAT 145

Wednesday, April 17, 2019 MAT 145

Try These! Wednesday, April 17, 2019 MAT 145

The Fundamental Theorem of Calculus (Part II) Let f be continuous on [a, b]. Then, where F is any antiderivative of f; that is, F ′(x) = f(x). Wednesday, April 17, 2019 MAT 145

The Fundamental Theorem of Calculus (Part II) Wednesday, April 17, 2019 MAT 145

The Fundamental Theorem of Calculus (Part II) Wednesday, April 17, 2019 MAT 145

The Fundamental Theorem of Calculus (Part II) Wednesday, April 17, 2019 MAT 145

The Fundamental Theorem of Calculus (Part I) For f continuous on [a, b], let the function g be Then g(x) is an antiderivative of f: Wednesday, April 17, 2019 MAT 145

Rates and Accumulations Water flows out of a tank at liters per second, t ≥ 0. (1) How much water left the tank during the first 2 minutes of drainage? (2) If the tank contained 6000 liters of water when the outflow began, how much water was in the tank after 5 minutes of drainage? (3) If the tank contained 6000 liters of water when the outflow began, how much time is required to drain the tank? (nearest second) Wednesday, April 17, 2019 MAT 145

Rates and Accumulations Water flows out of a tank at liters per second, t ≥ 0. How much water left the tank during the first 2 minutes of drainage? 886 2/3 liters of water left the tank during the first two minutes. Wednesday, April 17, 2019 MAT 145

Rates and Accumulations Water flows out of a tank at liters per second, t ≥ 0. (2) If the tank contained 6000 liters of water when the outflow began, how much water was in the tank after 5 minutes of drainage? There are 2519.230 liters lefts in the tank after 5 minutes. Wednesday, April 17, 2019 MAT 145

Rates and Accumulations Water flows out of a tank at liters per second, t ≥ 0. (3) If the tank contained 6000 liters of water when the outflow began, how much time is required to drain the tank? (nearest second) It will take about 432 seconds to drain the tank. Wednesday, April 17, 2019 MAT 145

Rates and Accumulations The velocity function (in meters per second) is given for a particle moving along a line. Calculate the displacement for the given time interval. (Note: Displacement is difference between starting position and ending position.) The displacement between t=0 and t=4 is 0 units. Wednesday, April 17, 2019 MAT 145