Basic Cal Quarter 4 Week 5 Illustrating Definite Integrals PDF

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Republic of the PhilippinesDepartment of EducationREGION IV-A CALABARZONSCHOOLS DIVISION OF BATANGASTUY SENIOR HIGH SCHOOLQuarter 4 Week 5SIMPLIFIED ACTIVITY SHEET IN BASIC CALCULUS Reference: Basic Calculus by: Carlene Perpetua P. Arceo, Ph.I. Learning Competency/iesa. Illustrate the definite integ...

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Republic of the Philippines

Department of Education REGION IV-A CALABARZON SCHOOLS DIVISION OF BATANGAS TUY SENIOR HIGH SCHOOL

Quarter 4 Week 5 SIMPLIFIED ACTIVITY SHEET IN BASIC CALCULUS Reference: Basic Calculus by: Carlene Perpetua P. Arceo, Ph.D. I.

Learning Competency/ies a. b.

II.

Illustrate the definite integral as the limit of the Reimann sums Illustrate the Fundamental Theorem of Calculus

Directions/ Instructions The following are some reminders in using this module: 1. Read the instruction carefully before doing each task. 2. Observe honesty and integrity in doing the tasks. 3. Finish the task at hand before proceeding to the next. 4. If you encounter any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator. 5. Always bear in mind that you are not alone. We hope that through this material, you will experience meaningful learning and gain deep understanding of the relevant competencies. You can do it!

III.

Objectives a. Illustrate the definite integral as the limit of the Reimann sums 1. Cognitive: Identify the formula using the limit of Reimann sums. 2. Affective: Appreciate other methods in finding the area of functions. 3. Cognitive: Illustrate the definite integral as the limit of Reimann sums b.

IV.

Illustrate the Fundamental Theorem of Calculus 1. Cognitive: Recognize the fundamental theorem of calculus 2. Affective: Apply critical thinking and analysis in performing the task 3. Psychomotor: Apply the fundamental theorem of calculus in evaluating the limit of functions.

Discussion

RIEMANN SUMS Throughout this lesson, we will assume that function f is positive (that is, the graph is above the x-axis), and continuous on the closed and bounded interval [a, b]. The goal of this lesson is to approximate the area of the region R bounded by y = f(x), x = a, x = b, and the x-axis. 1 Imagine we want to find the area under the graph of𝑓(𝑥) = 𝑥 2 between 𝑥 = 2 and 𝑥 = 6. 5

We first partition [a, b] regularly, that is, into congruent subintervals. Similar to the method of exhaustion, we fill R with rectangles of equal widths. The Riemann sum of f refers to the number equal to the combined area of these rectangles. Notice that as the number of rectangles increases, the Riemann sum approximation of the exact area of R becomes better and better. Of course, the Riemann sum depends on how we construct the rectangles and with how many rectangles we fill the region. We will discuss three basic types of Riemann sums: Left, Right, and Midpoint. PARTITION POINTS First, we discuss how to divide equally the interval [a, b] into n subintervals. To do this, we compute the step size

Next, we let x0 = a, and for each i = 1, 2,...,n, we set the ith intermediate point to be 𝑥𝑖 = 𝑎 + 𝑖∆𝑥. Clearly, the last point is

𝑥𝑛 = 𝑎 + 𝑛∆𝑥 = 𝑎 + 𝑛 (

𝑏−𝑎 𝑛

) = 𝑏. Please refer to the following table:

We call the collection of points Pn = {x0, x1,...,xn} a set of partition points of [a, b]. Note that to divide an interval into n subintervals, we need n + 1 partition points.

EXAMPLE 1: Find the step size ∆𝑥 and the partition points needed to divide the given interval into the given number of subintervals.

Assume that the interval [a, b] is already divided into n subintervals. We then cover the region with rectangles whose bases correspond to a subinterval. The three types of Riemann sum depend on the heights of the rectangles we are covering the region with. LEFT RIEMANN SUM The nth left Riemann sum Ln is the sum of the areas of the rectangles whose heights are the functional values of the left endpoints of each subinterval. For example, we consider the following illustration. We subdivide the interval into three subintervals corresponding to three rectangles. Since we are considering left endpoints, the height of the first rectangle is f(x0), the height of the second rectangle is f(x1), and the height of the third rectangle is f(x2).

2

Therefore, in this example, the 3rd left Riemann sum equals

In general, if [a, b] is subdivided into n intervals with partition points {x0, x1,...,xn}, then the nth left Riemann sum equals

We define the right and midpoint Riemann sums in a similar manner. RIGHT RIEMANN SUM The nth right Riemann sum Rn is the sum of the areas of the rectangles whose heights are the functional values of the right endpoints of each subinterval. For example, we consider the following illustration. We subdivide the interval into three subintervals corresponding to three rectangles. Since we are considering right endpoints, the height of the first rectangle is f(x1), the height of the second rectangle is f(x2), and the height of the third rectangle is f(x3).

Therefore, in this example, the 3rd right Riemann sum equals

In general, if [a, b] is subdivided into n intervals with partition points {x0, x1,...,xn}, then the nth right Riemann sum equals

MIDPOINT RIEMANN SUM The nth midpoint Riemann sum Mn is the sum of the areas of the rectangles whose heights are the functional values of the midpoints of the endpoints of each subinterval. For the sake of notation, we denote by mk the midpoint of two consecutive partition points xk.

3

We now consider the following illustration. We subdivide the interval into three subintervals corresponding to three rectangles. Since we are considering midpoints of the endpoints, the height of the first rectangle is f(m 1), the height of the second rectangle is f(m2), and the height of the third rectangle is f(m3).

Therefore, in this example, the 3rd midpoint Riemann sum equals

In general, if [a, b] is subdivided into n intervals with partition points {x0, x1,...,xn}, then the nth midpoint Riemann sum equals

EXAMPLE 2: Find the 4th left, right, and midpoint Riemann sums of the following functions with respect to a regular partitioning of the given intervals. a.

𝑓 (𝑥) = 𝑥2 𝑜𝑛 [0,1]

b. 𝑓 (𝑥) = sin 𝑥 𝑜𝑛 [0, 𝜋]

Solution: a.

First, note that ∆𝑥 =

1 3 5 7 {8 , 8 , 8 , }. 8

1−0 4

1

1 1 3

= . Hence, 𝑃4 = {0, , , , 1}. we then compute the midpoints of the partition points: 4 4 2 4

The 4th left Reimann sum equals

The 4th right Riemann sum equals

4

Lastly, the 4th midpoint Riemann sum equals

b.

First, note that ∆𝑥 =

𝜋 3𝜋 5𝜋 7𝜋 {8 , 8 , 8 , }. 8

1−0 4

𝜋

𝜋 𝜋

= 4 . Hence, 𝑃4 = {0, 4 , 2 ,

3𝜋 4

, 1}. we then compute the midpoints of the partition points:

The 4th left Riemann sum equals

The 4th right Riemann sum equals

Finally, the 4th midpoint Riemann sum equals

Lesson 3: Fundamental Theorem of Calculus We learned the Fundamental Theorem of Calculus (FTC), which from here forward will be referred to as the First Fundamental Theorem of Calculus, as in this section we develop a corresponding result that follows it. In particular, recall that the First FTC tells us that if f is a continuous function on [a,b] and F is any anti-derivative of 𝑓(𝑡ℎ𝑎𝑡 𝑖𝑠 , 𝐹 ′ = 𝑓), , then 𝑏

∫ 𝑓(𝑥)𝑑𝑥 = 𝐹(𝑏) − 𝐹(𝑎) 𝑎

This result, while taught early in elementary calculus courses, is actually a very deep result connecting the purely algebraic indefinite integral and the purely analytic (or geometric) definite integral. The second fundamental theorem of calculus holds for f a continuous function on an open interval 𝐼 and 𝑎 any point in 𝐼 , and states that if 𝐹 is defined by 𝑥

𝐹(𝑥) = ∫ 𝑓(𝑡)𝑑𝑡, 𝑎

Then

𝐹(𝑥) = 𝑓(𝑥) 𝑥

Example 1:Evaluate 𝑓(𝑥) = ∫0 √𝑡2 + 4 Solution: 𝑓 ′ (𝑥) = √𝑥2 + 4

Substitute x to the variable t to get the derivative of the function.

5

𝑥

Example 2: Evaluate ∫1 (𝑡 2 − 1)20𝑑𝑡 Solution: 𝑓 ′ (𝑥) = (𝑥 2 − 1)20 5

Example 3: Evaluate 𝑔(𝑥) = ∫𝑥 √𝑡 3 + 5 𝑑𝑡 Solution: 𝒙 𝒈(𝒙) = − ∫𝟓 √𝑡 3 + 5 𝒅𝒕 Interchange the upper and lower limit and the function becomes negative (Note the red font) 𝒈′ (𝒙) = −√𝒙𝟑 + 𝟓 V.

Substitute x to the variable t to get the derivative of the function

Activities/Instructions a. Illustrating Definite Integral as the limit of The Reimann Sums Directions: Express the following definite integrals as the limit of Reimann sums. Write the solutions and answers on the separate sheets of paper.

1.

2.

3.

4.

Let 𝑓(𝑥) = 𝑥 3be defined on [0, 1]. Find the left Riemann sum relative to the regular partitions P2, P3, P4. From these three approximations, could you guess what is the area of the region bounded by y = x 3 and the xaxis on [0, 1]? Let 𝑓(𝑥) = 3𝑥 − 𝑥 2 be defined on [0,2]. Find the right Riemann sum relative to the regular partitions P2, P3, P4. From these three approximations, could you guess what is the area of the region bounded by 𝑦 = 3𝑥 − 𝑥 2 and the x-axis on [0,2] Let 𝑓(𝑥) = √𝑥 be defined on [0, 1]. Find the midpoint Riemann sum relative to the regular partitions P2, P3, P4. From these three approximations, could you guess what is the area of the region bounded b 𝑦 = √𝑥 and the x-axis on [0, 1]? 𝜋 Let 𝑓(𝑥) = tan 𝑥 be defined on [0, ] . Find the left, right, and midpoint Riemann sums relative to the regular 4

partitions P2 and P3. (Do not forget to put your calculators to radian measure mode.) b. Illustrating Definite Integral using Fundamental Theorem of Calculus Directions: Find the derivative using fundamental theorem of calculus. F paper. Write the solutions and answers on the separate sheets of paper. 1. 2. 3. 4. 5. VI.

𝑥

𝑔(𝑥) = ∫1 (𝑡 6 + 5)5𝑑𝑡 𝑥 𝑓(𝑥) = ∫3 √𝑡 3 − 7 𝑑𝑡 𝑥 ℎ (𝑥) = ∫ cos(𝑡)8 𝑑𝑡 0 1

𝑓(𝑥) = ∫𝑥 𝑡 8 − 11 𝑑𝑡 2

ℎ(𝑥) = ∫𝑥 sin 𝑡 6 𝑡 𝑑𝑡

Reflection Write your insights about the lesson on a separate sheet of papers. I understand that_______________________________________________________________________________ ________________________________________________________________________________________________________ I realized that __________________________________________________________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________

PREPARED: ELSIE M. DE LOS REYES Math Teacher

NOTED: MA. VERLA AFRICA ALVARAN Principal II

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