Calculus I (MAT 145) Dr. Day Monday April 15, 2019 Chapter 5: Integrals and Anti-Derivartives 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” Monday, April 15, 2019 MAT 145

If we know a rate function . . . A particle moves along the x-axis. It’s velocity is given by v(t) = 2t2-3t+1 If we know that the particle is at location s = 3 at time t = 0, that is, that s(0) = 3, determine the position function s(t). What is the particle’s location at time t = 10? Monday, April 15, 2019 MAT 145

If we know a rate function . . . Snow begins falling at midnight at a rate of 1 inch of snow per hour. It stops snowing at 6 am, 6 hours later. Write an accumulation function S(t), to describe the total amount of snow that had fallen by time t, where 0 ≤ t ≤ 6 hrs. Monday, April 15, 2019 MAT 145

Accumulate, Accumulate, Accumulate! How much snow fell? Monday, April 15, 2019 MAT 145

Accumulate, Accumulate, Accumulate! How much snow fell? Monday, April 15, 2019 MAT 145

Accumulate, Accumulate, Accumulate! How much snow fell? Monday, April 15, 2019 MAT 145

Accumulate, Accumulate, Accumulate! How much snow fell? Monday, April 15, 2019 MAT 145

Accumulate, Accumulate, Accumulate! How much snow fell? Monday, April 15, 2019 MAT 145

Accumulate, Accumulate, Accumulate! How much snow fell? Monday, April 15, 2019 MAT 145

Accumulate, Accumulate, Accumulate! How much snow fell? Monday, April 15, 2019 MAT 145

Accumulate, Accumulate, Accumulate! How much snow fell? Monday, April 15, 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 Monday, April 15, 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. Monday, April 15, 2019 MAT 145

Monday, April 15, 2019 MAT 145

Approximating Area: Riemann Sums Riemann Sum Applet Monday, April 15, 2019 MAT 145

Monday, April 15, 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. Monday, April 15, 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 ? Monday, April 15, 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 Monday, April 15, 2019 MAT 145

Monday, April 15, 2019 MAT 145

Monday, April 15, 2019 MAT 145