1. C.1.5 – WORKING WITH DEFINITE INTEGRALS & FTC (PART 1) Calculus - Santowski 6/30/2016 1 Calculus - Santowski

2. (A) EVALUATING DEFINITE INTEGRALS Evaluate the following integrals: 6/30/2016 2 Calculus - Santowski

3. (B) DEFINITE INTEGRALS & DISCONTINUOUS FUNCTIONS Evaluate the following definite integrals: 6/30/2016 3 Calculus - Santowski

4. (B) DEFINITE INTEGRALS & DISCONTINUOUS FUNCTIONS Let and let (a) Carry out the integration. (b) Sketch the graphs of f(x) and g(x) (c) Where is f(x) continuous? Where is f(x) differentiable? Where is g(x) differentiable? 6/30/2016 4 Calculus - Santowski

5. (B) DEFINITE INTEGRALS & DISCONTINUOUS FUNCTIONS True or false & explain your answer: 6/30/2016 5 Calculus - Santowski

6. (C) APPLICATIONS OF DEFINITE INTEGRALS – MOTION PROBLEMS 1. An object starts at the origin and moves along the x-axis with a velocity v ( t ) = 10 t - t 2 for 0 < t < 10 (a) What is the position of the object at any time, t ? (b) What is the position of the object at the moment when it reaches its maximum velocity? (c) How far has the object traveled before it starts to decelerate? 6/30/2016 6 Calculus - Santowski

7. (C) APPLICATIONS OF DEFINITE INTEGRALS – MOTION PROBLEMS 1. For the velocity functions given below for a particle moving along a line, determine the distance traveled and displacement of the particle: (a) v ( t ) = 3 t – 5, 0 < t < 3 (b) v(t) = t 2 – 2 t – 8, 1 < t < 6 2. The acceleration function and the initial velocity are given for a particle moving along a line. Determine (a) the velocity at time t and (b) the distance traveled during the given time interval: (a) a ( t ) = t + 4, v (0) = 5, 0 < t < 10 (b) a ( t ) = 2 t + 3, v (0) = -4, 0 < t < 3 6/30/2016 7 Calculus - Santowski

8. (C) APPLICATIONS OF DEFINITE INTEGRALS – MOTION PROBLEMS Two cars, who are beside each other, leave an intersection at the same instant. They travel along the same road. Car A travels with a velocity of v ( t ) = t 2 – t – 6 m/s while Car B travels with a velocity of v ( t ) = 0.5 t + 2 m/s. (a) What are the initial velocities of the cars? (b) How far has each car gone after 4 seconds have elapsed? (c) When are the two cars beside each other again (i.e. when does the trailing car catch up to the leading car?) 6/30/2016 8 Calculus - Santowski

9. (D) AREA BETWEEN CURVES Find the area of the regions bounded: (a) above f(x) = x + 2 and below g(x) = x 2 (b) by f(x) = 4x and g(x) = x 3 from x = -2 to x = 2 (c) Evaluate and interpret the result in terms of areas. Then find the area between the graph of f(x) = x 2 – 2x and the x-axis from x = -1 to x = 3 6/30/2016 9 Calculus - Santowski

10. (C) AREA BETWEEN CURVES Sketch the curve of f(x) = x 3 – x 4 between x = 0 and x = 1. (a) Draw a vertical line at x = k such that the region between the curve and axis is divided into 2 regions of equal area. Determine the value of k. (b) Draw a horizontal line at y = h such that the region between the curve and axis is divided into 2 regions of equal area. Estimate the value of h. Justify your estimation. 6/30/2016 10 Calculus - Santowski

11. (D) HOMEWORK Worksheets (TBA) 6/30/2016 11 Calculus - Santowski