C.1.3b - Area & Definite Integrals: An Algebraic Perspective Calculus - Santowski 1/7/2016Calculus - Santowski 1

Fast Five 1/7/2016Calculus - Santowski2  (1) 0.25[f(1) + f(1.25) + f(1.5) + f(1.75)] if f(x) = x 2 + x - 1  (2) Illustrate Q1 with a diagram, showing all relevant details

(A) Review 1/7/2016Calculus - Santowski3  We will continue to move onto a second type of integral  the definite integral  Last lesson, we estimated the area under a curve by constructing/drawing rectangles under the curve  Today, we will focus on doing the same process, but from an algebraic perspective, using summations

(B) Summations, ∑ 1/7/2016Calculus - Santowski4  A series is the sum of a sequence (where a sequence is simply a list of numbers)  Ex: the sequence 2,4,6,8,10,12, …..2n has a an associated sum, written as:  The sum … + 2n can also be written as 2( … + n) or:

(C) Working with Summations 1/7/2016Calculus - Santowski5  Ex 1. You are given the following series:  List the first 7 terms of each series

(C) Working with Summations 1/7/2016Calculus - Santowski6  Ex 1I. You are given the following series:  Evaluate each series

(C) Working with Summations 1/7/2016Calculus - Santowski7  Ex 1. You are given the following series:  List the first 10 terms of each series

(C) Working with Summations 1/7/2016Calculus - Santowski8  Ex 1. You are given the following series:  Evaluate each series

(B) Summations, ∑ 1/7/2016Calculus - Santowski9  Here are “sum” important summation formulas:  (I) sum of the natural numbers  ( ….n)  (II) sum of squares  ( ….+n 2 )  (III) sum of cubes  ( ….n 3 )

(E) Working With Summations - GDC  Now, to save all the tedious algebra (YEAHH!!!), let’s use the TI-89 to do sums  First, let’s confirm our summation formulas for i, i 2 & i 3 and get acquainted with the required syntax 1/7/2016Calculus - Santowski10

(E) Working With Summations - GDC  So let’s revisit our previous example of  And our 4th example of 1/7/2016Calculus - Santowski11

(F) Applying Summations 1/7/2016Calculus - Santowski12  So what do we need summations for?  Let’s connect this algebra skill to determining the area under curves ==> after all, we are simply summing areas of individual rectangles to estimate an area under a curve

PART 2 - The Area Problem 1/7/2016Calculus - Santowski13  Let’s work with a simple quadratic function, f(x) = x and use a specific interval of [0,3]  Now we wish to estimate the area under this curve

(A) The Area Problem – Rectangular Approximation Method (RAM) 1/7/2016Calculus - Santowski14  To estimate the area under the curve, we will divide the are into simple rectangles as we can easily find the area of rectangles  A = l × w  Each rectangle will have a width of  x which we calculate as (b – a)/n where b represents the higher bound on the area (i.e. x = 3) and a represents the lower bound on the area (i.e. x = 0) and n represents the number of rectangles we want to construct  The height of each rectangle is then simply calculated using the function equation  Then the total area (as an estimate) is determined as we sum the areas of the numerous rectangles we have created under the curve  A T = A 1 + A 2 + A 3 + ….. + A n  We can visualize the process on the next slide

(A) The Area Problem – Rectangular Approximation Method (RRAM) 1/7/2016Calculus - Santowski15  We have chosen to draw 6 rectangles on the interval [0,3]  A 1 = ½ × f(½) =  A 2 = ½ × f(1) = 1.5  A 3 = ½ × f(1½) =  A 4 = ½ × f(2) = 3  A 5 = ½ × f(2½) =  A 6 = ½ × f(3) = 5.5  A T = square units  So our estimate is which is obviously an overestimate

(B) Sums & Rectangular Approximation Method (RRAM)  So let’s apply our summation formulas:  Each rectangle’s area is f(x i )  x where f(x) = x   x = 0.5 and x i = 0 +  xi  Therefore the area of 6 rectangles is given by 1/7/2016Calculus - Santowski16

(C) The Area Problem – Rectangular Approximation Method (LRAM) 1/7/2016Calculus - Santowski17  In our previous slide, we used 6 rectangles which were constructed using a “right end point” (realize that both the use of 6 rectangles and the right end point are arbitrary!)  in an increasing function like f(x) = x this creates an over-estimate of the area under the curve  So let’s change from the right end point to the left end point and see what happens

(C) The Area Problem – Rectangular Approximation Method (LRAM) 1/7/2016Calculus - Santowski18  We have chosen to draw 6 rectangles on the interval [0,3]  A 1 = ½ × f(0) = 1  A 2 = ½ × f(½) =  A 3 = ½ × f(1) = 1.5  A 4 = ½ × f(1½) =  A 5 = ½ × f(2) = 3  A 6 = ½ × f(2½) =  A T = square units  So our estimate is which is obviously an under-estimate

(D) Sums & Rectangular Approximation Method (LRAM)  So let’s apply our summation formulas:  Each rectangle’s area is f(x i )  x where f(x) = x   x = 0.5 and x i = 0 +  xi  Therefore the area of 6 rectangles is given by: (Notice change in i???) 1/7/2016Calculus - Santowski19

(E) The Area Problem – Rectangular Approximation Method (MRAM) 1/7/2016Calculus - Santowski20  So our “left end point” method (called a left hand Riemann sum or LRAM) gives us an underestimate (in this example)  Our “right end point” method (called a right handed Riemann sum or RRAM) gives us an overestimate (in this example)  We can adjust our strategy in a variety of ways  one is by adjusting the “end point”  why not simply use a “midpoint” in each interval and get a mix of over- and under-estimates?  see next slide

(E) The Area Problem – Rectangular Approximation Method (MRAM) 1/7/2016Calculus - Santowski21  We have chosen to draw 6 rectangles on the interval [0,3]  A 1 = ½ × f(¼) =  A 2 = ½ × f (¾) =  A 3 = ½ × f(1¼) =  A 4 = ½ × f(1¾) =  A 5 = ½ × f(2¼) =  A 6 = ½ × f(2¾) =  A T = square units which is a more accurate estimate (15 is the exact answer)

(F) Sums & Rectangular Approximation Method (MRAM)  So let’s apply our summation formulas:  Each rectangle’s area is f(x i )  x where f(x) = x   x = 0.5 and x i =  xi  Therefore the area of 6 rectangles is given by: (Notice change in i??? and the x i expression???) 1/7/2016Calculus - Santowski22

(G) The Area Problem – Expanding our Example 1/7/2016Calculus - Santowski23  Now back to our left and right Riemann sums and our original example  how can we increase the accuracy of our estimate?  We simply increase the number of rectangles that we construct under the curve  Initially we chose 6, now let’s choose a few more … say 12, 60, and 300 ….  But first, we need to generalize our specific formula!

(G) The Area Problem – Expanding our Example 1/7/2016Calculus - Santowski24  So we have  Now,  x = (b-a)/n = (3 - 0)/n = 3/n  And f(x i ) = f(a +  xi) = f(0 + 3i/n) = f(3i/n)  So, we have to work with the generalized formula

(G) The Area Problem – Expanding our Example  Does this generalized formula work?  Well, test it with n = 6 as before! 1/7/2016Calculus - Santowski25

(G) The Area Problem – Expanding our Example (RRAM) # of rectanglesArea estimate /7/2016Calculus - Santowski26

(G) The Area Problem – Expanding our Example # of rectanglesArea estimate /7/2016Calculus - Santowski27

(G) The Area Problem – Expanding our Example (LRAM) # of rectanglesArea estimate /7/2016Calculus - Santowski28

(G) The Area Problem – Expanding our Example (LRAM) # of rectanglesArea estimate /7/2016Calculus - Santowski29

(H) The Area Problem - Conclusion  So our exact area seems to be “sandwiched” between and !!!  So, if increasing the number of rectangles increases the accuracy, the question that needs to be asked is ….. how many rectangles should be used???  The answer is ….. why not use an infinite number of rectangles!!  so now we are back into LIMITS!!  So, the exact area between the curve and the x-axis can be determined by evaluating the following limit: 1/7/2016Calculus - Santowski30

(H) The Area Problem - Conclusion  So let’s verify the example using the GDC and limits: 1/7/2016Calculus - Santowski31

(I) The Area Problem – Further Examples 1/7/2016Calculus - Santowski32  (i) Determine the area between the curve f(x) = x 3 – 5x 2 + 6x + 5 and the x-axis on [0,4] if we (a) construct 20 rectangles or (b) if we want the exact area  (ii) Determine the exact area between the curve f(x) = x 2 – 4 and the x-axis on [0,2] if we (a) construct 30 rectangles or (b) if we want the exact area  (iii) Determine the exact area between the curve f(x) = x 2 – 2 and the x-axis on [0,2] if we (a) construct 10 rectangles or (b) if we want the exact area

(J) Homework 1/7/2016Calculus - Santowski33  Homework to reinforce these concepts from this second part of our lesson:  Handout, Stewart, Calculus - A First Course, 1989, Exercise 10.4, p474-5, Q3,4,6a

Internet Links 1/7/2016Calculus - Santowski34  Calculus I (Math 2413) - Integrals - Area Problem from Paul Dawkins Calculus I (Math 2413) - Integrals - Area Problem from Paul Dawkins  Integration Concepts from Visual Calculus Integration Concepts from Visual Calculus  Areas and Riemann Sums from P.K. Ving - Calculus I - Problems and Solutions Areas and Riemann Sums from P.K. Ving - Calculus I - Problems and Solutions