v(t) v(t) Assignment & Notes 5.2: Intro to Integrals Due Date: Friday, 1/10

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Assignment & Notes 5.2: Intro to Integrals 1. The velocity function (in miles and hours) for Ms. Hardtke s Christmas drive to see her family is shown at the right. Find the total distance Ms.

1 Assignment & Notes 5.2: Intro to Integrals 1. The velocity function (in miles and hours) for Ms. Hardtke s Christmas drive to see her family is shown at the right. Find the total distance Ms. H travelled between ½ hour and 3 hours. AB Calculus - Hardtke Name Due Date: Friday, 1/10 70 miles v(t) Can you find the total distance Ms. H travelled between 0 hours and 3.5 hours? hours 3. How might you approximate the distance travelled between 0 hours and 2 hours for this velocity function? miles v(t) hours To approximate the area under a curve (or accumulated distance from a velocity function, we can divide the region into rectangles and find the sum of the areas. Symbol for Riemann Sum to add a large number of thin rectangles to approximate such an area we write: If we use to add an number of very thin rectangles this becomes the area By definition, an infinite Riemann sum is a Definite Integral and we write:

Example 1A 1B: Approximate the area under the curve y = x 2 from x = 0 to x = 1 using 8 "Left-Endpoint rectangles" and then by using 8 Right-Endpoint Rectangles.

2 Example 1A 1B: Approximate the area under the curve y = x 2 from x = 0 to x = 1 using 8 "Left-Endpoint rectangles" and then by using 8 Right-Endpoint Rectangles. In symbols: Find L 8 and R 8 Example 2: Approximate the area under the curve f(x) = e -x from x = 0 to x = 2 by using 4 "Midpoint rectangles" In symbols: For f(x) = e x, find M 4 When we are given tabular velocity data, we can only estimate total distance travelled by using a sum of rectangles. Example 3A & B: Estimate total distance by finding: a) L 3 and then by finding b) R 3. Time (hours) Velocity (mi/h)

3 Assignment 5.2: Intro to Integrals AB Calculus - Hardtke Name Due Date: Friday 1/10 Show your set-up using a Riemann sum for each problem. You may use a calculator for computations if you wish. 1. Approximate by sketching a graph and finding: A. L 3 B. R 3 C. M 3 D. Can you tell which answer(s) above are over-estimates or under-estimates of the exact area? How do you know? E. On your TI-89, note the integral symbol above the 7 key. Calculate. F. What is the simplest of the antiderivatives of f(x) = x^3? G. Given F(x) = ¼ x 4, find F(3) F(0). Over

4 2. Approximate by sketching a graph and finding: A. L 3 B. R 3 C. Can you tell which answer(s) above are over-estimates or under-estimates of the exact area? How do you know? D. On your TI-89, in radian mode as always, calculate. E. What is the simplest of all antiderivatives of f(x) = cos x? F. Given F(x) = sin x, find F( ) F(0). 3. For the tabular velocity data below, estimate the total distance by finding: a) L 5 and then by finding b) R 5. c) Do you know which is the better estimate? Why or why not? Time (hours) Velocity (mi/h)

5 AB Calculus - Hardtke Assignment 5.2: Intro to Integrals SOLUTION KEY Show your set-up for each problem. You may use a calculator for computations if you wish. 1. Approximate by sketching a graph and finding: A. L3 B. R3 C. M3 D. Can you tell which answer(s) above are over-estimates or under-estimates of the exact area? How do you know? E. On your TI-89, note the integral symbol above the 7 key. Calculate. F. What is the simplest of the antiderivatives of f(x) = x^3? 4 G. Given F(x) = ¼ x, find F(3) F(0).

6 2. Approximate by sketching a graph and finding: A. L3 B. R3 C. Can you tell which answer(s) above are over-estimates or under-estimates of the exact area? How do you know? D. On your TI-89, in radian mode as always, calculate. E. What is the simplest of all antiderivatives of f(x) = cos x? F. Given F(x) = sin x, find F( ) F(0). 3. For the tabular velocity data below, estimate the total distance by finding: a) L5 and then by finding b) R5. c) Do you know which is the better estimate? Why or why not? Time (hours) Velocity (mi/h)

Science One Integral Calculus

Science One Integral Calculus January 018 Happy New Year! Differential Calculus central idea: The Derivative What is the derivative f (x) of a function f(x)? Differential Calculus central idea: The Derivative