Limit Definition of the Definite Integral

AP® Calculus AB Lesson Plan Limit Definition of the Definite Integral

Duration NOTES One 90-minute class period Write or type in this area.

Resources

1. Presentation 2. Worksheet 3. Homework

AP® Calculus AB Worksheet AP® Calculus AB Homework Limit Definition of the Definite Integral Limit Definition of the Definite Integral Limit Deﬁnition of the Deﬁnite Integral

1. Identify Δ and , then express the following Riemann sums as definite integrals. Learning Outcomes: I can interpret the definite integral as the limit of a Riemann sum. (a) lim 2 2 3 3 (e) lim 5 5 3 ® AP Calculus AB →∞ =1 − + →∞ =1 + Using a Right Riemann Sum (b) lim 4 4 5 5 (f) lim 3 =1 = − →∞ →∞ 1 Lesson Plan: Limit Definition 3 (c) lim 1 2 2 (g) lim cos 4 2 2 =1 + =1 + →∞ →∞ of the Definite Integral 3 (d) lim 1 2 2 (h) lim 16 ln 4 + 2 →∞ =1 →∞ =1 1. Write an expression for the number of rectangles. 2. Use the limit definition of the definite integral to evaluate the following integrals. 3 2 (a) 4 (c) 1 2 1 1 ( − ) 2. Write an expression for the width of each rectangle. ∫ −∫ 4 3 (b) 3 4 (d) 3 1 0 ( + ) 1 ( + ) 3. Write an expression for the height of each rectangle. ∫ ∫

4. Write an expression for the area of each rectangle.

5. Write an expression for the sum of the areas of all the rectangles.

Definite Integral as the Limit of a Riemann Sum If f is a continuous function defined on a, b , and if: [ ] b a • a, b is divided into n equal subintervals of width ∆x = − [ ] n • xk = a + k∆x is the right endpoint of subinterval k. then the definite integral of f from a to b is the number

b f x dx = Objectives of Lesson( ) a • To interpret a definite integral as the limit of a Riemann

© Marco Learning, LLC. All Rights Reserved. Advanced Placement® and AP® are trademarks © Marco Learning, LLC. All Rights Reserved. Advanced Placement® and AP® are trademarks Sum registered by the College Board, which is not affiliated with, and does not endorse, this product. Visit www.marcolearning.com for additional resources. registered by the College Board, which is not affiliated with, and does not endorse, this product. Visit www.marcolearning.com for additional resources. • To be able to represent a definite integral as the limit of a Riemann Sum • To be able to evaluate a definite integral using properties of limits and summations

College Board Objectives from the 2019–20 CED • Mathematical Practices—Practice 1: Implementing Mathematical Processes • Mathematical Practices—Practice 1F: Determine expressions and values using mathematical procedures and rules. • Mathematical Practices—Practice 2: Connecting Representations • Mathematical Practices—Practice 2C: Identify a re- expression of mathematical information • presented in a given representation. • Mathematical Practices—Practice 4: Communication and Notation—Use correct notation, language, and mathematical conventions to communicate results or solutions. • Mathematical Practices—Practice 4C: Use appropriate mathematical symbols and notation. • Learning Objective LIM-5.B: Interpret the limiting case of the Riemann Sum as a definite integral. • Learning Objective LIM-5.C: Represent the limiting case of the Riemann Sum as a definite integral. • Prior Knowledge: Students should be able to compute the value of a left-, right-, and midpoint Riemann Sum from work in previous lessons.

© Marco Learning, LLC. All Rights Reserved. Advanced Placement® and AP® are trademarks registered by the College Board, which is not affiliated with, and does not endorse, this product. Visit www.marcolearning.com for additional resources. AP® Calculus AB Lesson Plan Limit Definition of the Definite Integral

How to Use This Lesson Plan NOTES The slide presentation is meant to be used during a whole group Write or type in this area. discussion. Students fill in the information on the worksheet as the lesson progresses. Following the presentation, debrief with students and assign practice exercises for homework.

Instructions 1. Whole Group Discussion (slides 1–4). Slides 1–2 are meant to help develop the definition of the definite integral. The first example shows a right-hand Riemann Sum. Discuss the meaning of the notation depicted in the graph, and emphasize that, although there appear to be 10 rectangles, the graph is meant to be generalized to n rectangles (subintervals). Students may struggle to understand the meaning of k in this context, so be sure to discuss the meaning of x_k as the kth rectangle (subinterval) and identify k as a “counter.” 2. Formative Assessment (slides 5–6). Check for understanding using voting cards or think-pair-share. 3. Whole Group Discussion (slides 7–12). Work through the examples on slides 9 and 11 together, emphasizing the importance of proper mathematical notation as you go. Students are not required to memorize the summation properties and formulas. They should, however, highlight these properties and formulas in their notes, so they can refer to them when working through the homework exercises. 4. Formative Assessment (slide 13). Have students work independently to solve the problem on slide 13. 5. Follow-Up Questions. Debrief with a whole-group discussion: a. How are Riemann Sums used to approximate the area under a curve on a given interval? b. How are limits used to find the exact area under a curve on a given interval? c. The examples shown involved using a right-hand Riemann Sum. Would this technique work for a left- hand Riemann Sum? A midpoint Riemann Sum?

2 AP® Calculus AB

Lesson Plan: Limit Deﬁnition of the Deﬁnite Integral Limit of a Riemann Sum

Write an expression for each of the following quantities:

1. the number of rectangles. 4. the area of each rectangle. 2. the width of each rectangle. 5. the total area of all rectangles. 3. the height of each rectangle.

www.marcolearning.com 1/14 Limit of a Riemann Sum

Write an expression for each of the following quantities:

1. the number of rectangles. n 4. the area of each rectangle. f(xk )∆x ∆ b−a 5. the total area of all rectangles. 2. the width of each rectangle. x = n Ín f(xk )∆x 3. the height of each rectangle. f(xk ) k=1

www.marcolearning.com 1/14 Limit of a Riemann Sum

x1 = a + ∆x

x2 =

x3 = . .

xk =

www.marcolearning.com 2/14 Limit of a Riemann Sum

x1 = a + ∆x

x2 = a + 2∆x

x3 = a + 3∆x . .

xk = a + k∆x

www.marcolearning.com 2/14 Deﬁnite Integral as the Limit of a Riemann Sum

Definition of Definite Integral If f is a continuous function defined on [a, b], and if:

[ ] ∆ b−a • a, b is divided into n equal subintervals of width x = n ,

• and if xk = a + k∆x is the right endpoint of subinterval k,

then the deﬁnite integral of f from a to b is the number

b ¹ Õn f(x) dx = lim f(xk ) ∆x n→∞ a k=1

www.marcolearning.com 3/14 Deﬁnite Integral as the Limit of a Riemann Sum

Let n be the number of subintervals. The exact area under the curve is given by the limit: n Í 5 5k 2 lim n n + 2 n→∞ k=1

n Í 5k 2 ∆ lim n + 2 x n→∞ k=1 which is exactly equal to:

∫5 (x2 + 2) dx 0

www.marcolearning.com 4/14 Deﬁnite Integral as the Limit of a Riemann Sum

∫5 Which of the following limits is equal to x4 dx? 3

n 4 Í k 1 (A) lim 3 + n n n→∞ k=1

n 4 Í k 2 (B) lim 3 + n n n→∞ k=1

n 4 Í 2k 1 (C) lim 3 + n n n→∞ k=1

n 4 Í 2k 2 (D) lim 3 + n n n→∞ k=1

www.marcolearning.com 5/14 Deﬁnite Integral as the Limit of a Riemann Sum

∫5 Which of the following limits is equal to x4 dx? 3

n 4 Í k 1 (A) lim 3 + n n n→∞ k=1

n 4 Í k 2 (B) lim 3 + n n n→∞ k=1

n 4 Í 2k 1 (C) lim 3 + n n n→∞ k=1

n 4 Í 2k 2 (D) lim 3 + n n n→∞ k=1

www.marcolearning.com 5/14 Deﬁnite Integral as the Limit of a Riemann Sum

∫5 Which of the following limits is equal to (4 − 2x) dx? 2 n Í − k 1 (A) lim 4 2 2 + n n n→∞ k=1

n Í − 3k 1 (B) lim 4 2 2 + n n n→∞ k=1

n Í − k 3 (C) lim 4 2 2 + n n n→∞ k=1

n Í − 3k 3 (D) lim 4 2 2 + n n n→∞ k=1

www.marcolearning.com 6/14 Deﬁnite Integral as the Limit of a Riemann Sum

∫5 Which of the following limits is equal to (4 − 2x) dx? 2 n Í − k 1 (A) lim 4 2 2 + n n n→∞ k=1

n Í − 3k 1 (B) lim 4 2 2 + n n n→∞ k=1

n Í − k 3 (C) lim 4 2 2 + n n n→∞ k=1

n 4 Í − 3k 3 (D) lim 4 2 2 + n n n→∞ k=1

www.marcolearning.com 6/14 Summation Properties

Recall the following summation properties:

n 1. Í c = k=1

Ín 2. cak = k=1

Ín 3. (ak ± bk ) = k=1

www.marcolearning.com 7/14 Summation Properties

Recall the following summation properties:

n 1. Í c = nc k=1

Ín Ín 2. cak = c ak k=1 k=1

Ín Ín Ín 3. (ak ± bk ) = ak ± bk k=1 k=1 k=1

www.marcolearning.com 7/14 Summation Formulas

Recall the following summation formulas:

n n(n + 1) 1. Í k = k=1 2

n n(n + 1)(2n + 1) 2. Í k2 = k=1 6

n n2(n + 1)2 3. Í k3 = k=1 4

www.marcolearning.com 8/14 Limit Deﬁnition of Deﬁnite Integral

∫ 5 (3x + 1) dx Use the deﬁnition of deﬁnite integral to evaluate 3 .

First, we write an expression for ∆x.

∆x =

Second, we write an expression for xk .

xk =

Third, we write the limit deﬁnition of the deﬁnite integral.

¹ 5 (3x + 1) dx = 3 www.marcolearning.com 9/14 Limit Deﬁnition of Deﬁnite Integral

∫ 5 (3x + 1) dx Use the deﬁnition of deﬁnite integral to evaluate 3 .

First, we write an expression for ∆x. 2 ∆x = n

Second, we write an expression for xk . 2k xk = 3 + n Third, we write the limit deﬁnition of the deﬁnite integral.

¹ 5 Õn (3x + 1) dx = lim [3(xk ) + 1] ∆x n→∞ 3 k=1 www.marcolearning.com 9/14 Limit Deﬁnition of Deﬁnite Integral

¹ 5 Õn (3x + 1) dx = lim [3(xk ) + 1] ∆x n→∞ 3 k=1 n Õ 2k 2 = lim 3 3 + + 1 n→∞ n n k=1 n Õ 20 12k = lim + n→∞ n n2 k=1 " n n # 20 Õ 12 Õ = lim 1 + k n→∞ n n2 k=1 k=1 20 12 n(n + 1) = lim · n + · n→∞ n n2 2 = 20 + 6 = 26 www.marcolearning.com 10/14 Limit Deﬁnition of Deﬁnite Integral

∫ 4 (2x2 + 3) dx Use the deﬁnition of deﬁnite integral to evaluate 0 .

First, we write an expression for ∆x.

∆x =

Second, we write an expression for xk .

xk =

Third, we write the limit deﬁnition of the deﬁnite integral.

¹ 4 (2x2 + 3) dx = 0 www.marcolearning.com 11/14 Limit Deﬁnition of Deﬁnite Integral

∫ 4 (2x2 + 3) dx Use the deﬁnition of deﬁnite integral to evaluate 0 .

First, we write an expression for ∆x. 4 ∆x = n

Second, we write an expression for xk . 4k xk = n Third, we write the limit deﬁnition of the deﬁnite integral.

¹ 4 n 2 Õ 2 (2x + 3) dx = lim [2(xk ) + 3] ∆x n→∞ 0 k=1 www.marcolearning.com 11/14 Limit Deﬁnition of Deﬁnite Integral

¹ 4 n 2 Õ 2 (2x + 3) dx = lim [2(xk ) + 3] ∆x n→∞ 0 k=1 n " # Õ 4k 2 4 = lim 2 + 3 n→∞ n n k=1 n Õ 128 12 = lim k2 + n→∞ n3 n k=1 " n n # 128 Õ 12 Õ = lim k2 + 1 n→∞ n3 n k=1 k=1 128 n(n + 1)(2n + 1) 12 = lim · + · n n→∞ n3 6 n 128 164 = + 12 = 3 3 www.marcolearning.com 12/14 Limit Deﬁnition of Deﬁnite Integral

∫ 5 (8x − x2) dx Use the deﬁnition of deﬁnite integral to evaluate 2 .

www.marcolearning.com 13/14 Limit Deﬁnition of Deﬁnite Integral

∫ 5 (8x − x2) dx Use the deﬁnition of deﬁnite integral to evaluate 2 . 45

www.marcolearning.com 13/14 Credits

Visit www.marcolearning.com for additional learning resources.

Advanced Placement® and AP® are trademarks registered by the College Board, which is not aﬃliated with, and does not endorse, this product.

www.marcolearning.com 14/14 AP® Calculus AB Worksheet Limit Definition of the Definite Integral Limit Deﬁnition of the Deﬁnite Integral

Learning Outcomes: I can interpret the deﬁnite integral as the limit of a Riemann sum.

Using a Right Riemann Sum

1. Write an expression for the number of rectangles.

2. Write an expression for the width of each rectangle.

3. Write an expression for the height of each rectangle.

4. Write an expression for the area of each rectangle.

5. Write an expression for the sum of the areas of all the rectangles.

b f x dx = ( ) a

© Marco Learning, LLC. All Rights Reserved. Advanced Placement® and AP® are trademarks registered by the College Board, which is not affiliated with, and does not endorse, this product. Visit www.marcolearning.com for additional resources. AP® Calculus AB Worksheet Limit Definition of the Definite Integral

Let n be the number of subintervals. The exact area under the curve is given by the limit:

n lim 5 5k 2 + 2 n n n →∞ k=1 n lim 5k 2 + 2 ∆x n n →∞ k=1 which is exactly equal to:

5 6. Write a limit that is equal to x4 dx. 3 ∫

5 7. Write a limit that is equal to 4 2x dx. 2 ( − ) ∫

Summation Properies

n 1. c = k=1 n 2. cak = k=1 n 3. ak bk = k=1( ± ) Summation Formulas

n 1. k = k=1 n 2. k2 = k=1 n 3. k3 = k=1

2 AP® Calculus AB Worksheet Limit Definition of the Definite Integral

5 8. Use the deﬁnition of deﬁnite integral to evaluate 3x + 1 dx. 3 ( ) First, we write an expression for ∆x. ∫

∆x =

Second, we write an expression for xk.

xk =

Third, we write the limit deﬁnition of the deﬁnite integral.

5 3x + 1 dx = ( ) 3

3 AP® Calculus AB Worksheet Limit Definition of the Definite Integral

4 9. Use the deﬁnition of deﬁnite integral to evaluate 2x2 + 3 dx. 0 ( ) First, we write an expression for ∆x. ∫

∆x =

Second, we write an expression for xk.

xk =

Third, we write the limit deﬁnition of the deﬁnite integral.

4 2x2 + 3 dx = ( ) 0

4 AP® Calculus AB Worksheet Limit Definition of the Definite Integral

5 10. Use the deﬁnition of deﬁnite integral to evaluate 8x x2 dx. 2 ( − ) ∫

5 AP® Calculus AB Homework Limit Definition of the Definite Integral

1. Identify Δ and , then express the following Riemann sums as deﬁnite integrals.

(a) lim 2 2 3 3 (e) lim 5 5 3 →∞ =1 − + →∞ =1 + (b) lim 4 4 5 5 (f) lim 3 =1 = − →∞ →∞ 1 3 (c) lim 1 2 2 (g) lim cos 4 2 2 =1 + =1 + →∞ →∞ 3 (d) lim 1 2 2 (h) lim 16 ln 4 + 2 →∞ =1 →∞ =1 2. Use the limit deﬁnition of the deﬁnite integral to evaluate the following integrals.

3 2 (a) 4 (c) 1 2 1 1 ( − ) ∫ −∫ 4 3 (b) 3 4 (d) 3 1 0 ( + ) 1 ( + ) ∫ ∫