Today s Agenda Upcoming Homework Section 5.1: Areas and Distances Section 5.2: The Definite Integral Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Wed., 18 November 2015 1 / 13
Upcoming Homework WeBWorK HW 22: Section 5.1, due 11/20/2015. Study for Exam 3 on 11/23/2015. WeBWorK HW 23: Section 5.2, due 11/25/2015. Written HW M: Section 5.1, #4. Section 5.2, #10,16,18,30,34,44. Due 11/30/2015. Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Wed., 18 November 2015 2 / 13
Suggestions for studying for Test 3 Topics covered on Test 3: Integrals in terms of areas using basic geometry, absolute maxima and minima, Mean Value Theorem, inflection points, right and left Riemann sums, properties of definite integrals, optimization problems, antiderivatives, concave up/concave down, intervals of increase/decrease, distance/velocity/acceleration problems. 1 Take the Old Exam 3 found at https://math.asu.edu/ first-year-math/mat-265-calculus-engineers-i. Consider timing yourself while you take the practice exam. 2 Do the Exam 3 review found at the same link. 3 Suggested practice problems from the textbook: 1 Chapter 4 Review, pages 254-256: Exercises (not concept check or true-false) #1-4, 9-14, 36-41, 51-58. 2 Section 4.5, pages 238-239: #7,8,11,12,14. 3 Section 4.5, Example 1, page 232. 4 Section 5.2, pages 279-281: #1-3, 31-36, 38-42. Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Wed., 18 November 2015 3 / 13
Section 5.1 When we are evaluating sums whose terms follow a pattern, it is useful to use sigma notation. Sigma Notation Suppose we are calculating the finite sum a 1 + a 2 + + a n. This sum is represented in sigma notation by a k. k=1 As an example, we write the sum of the first n positive integers as 1 + 2 + + n = k. k=1 Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Wed., 18 November 2015 4 / 13
Section 5.1 Example problem: Find a formula that approximates the area under the curve y = 4 x 2 using n rectangles of equal base length. Make sure the formula is in terms of only n, and use sigma notation. (You may be wondering why we need to do all this work. One application of this is in computer programming: the area under a curve can be calculated numerically by using these sum formulas and large values of n.) Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Wed., 18 November 2015 5 / 13
Section 5.1 We find that the area under the curve y = 4 x 2, approximated by a right-hand sum over n rectangles of equal base length is 32 64 3 32 3n 2. Obviously, the best approximation is reached by letting n, so we take the limit ( lim 32 64 n 3 32 ) 3n 2 = 32 64 3 = 32 3. This gives us the exact area under the curve. Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Wed., 18 November 2015 6 / 13
Section 5.1 We will now tackle the distance problem. Suppose we have a particle that is moving along a straight line with velocity given by the formula v(t) = 10t 3. Find the distance between the particle s starting point (at t = 0) and its final point (at t = 3). You may use the following formulas if necessary: i = i=1 n(n + 1) 2 i 2 = i=1 n(n + 1)(2n + 1) 6 ( n(n + 1) i 3 = 2 i=1 ) 2 Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Wed., 18 November 2015 7 / 13
Section 5.2 On Monday, we briefly mentioned that the partition of the interval over which we are calculating the area does not have to be evenly divided. We also mentioned many different ways that we could assign the heights of the rectangles we are using for our approximation. In general, given the interval [a, b] over which we wish to calculate the area, we can choose a partition of [a, b] by choosing partition points x 0, x 1,..., x n 1, x n as follows: a = x 0 < x 1 < < x n 1 < x n = b. We use the notation x i = x i x i 1 to denote the length of each subinterval of the partition. Then we may choose xi [x i 1, x i ] as a sample point that we will use to calculate the height. Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Wed., 18 November 2015 8 / 13
Section 5.2 Once we have chosen our partition and sample points, we represent our approximation of the area by f (xi ) x i, i=1 or in other words, the area is the sum of the rectangles formed by taking our partition as the base and the function value at our sample points as the height. This sum is known as a Riemann sum. Note that if the function value is negative, we assign a negative value to the area of the rectangle. By doing this, it is possible to calculate the area between the curve and the x-axis for functions that cross the x-axis (or functions that lie completely below the x-axis). Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Wed., 18 November 2015 9 / 13
Section 5.2 We are now in a position to define the definite integral. Definition: Definite Integral If f is a function defined on [a, b], the definite integral of f from a to b is the number b a f (x) dx = lim (max x i ) 0 f (xi ) x i, provided that this limit exists. If it does exist, we say that f is integrable on [a, b]. i=1 The symbol dx can be thought of as representing an infinitesimally small base length for the approximating rectangles. Don t forget the dx when you are writing formulas for integrals! Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Wed., 18 November 2015 10 / 13
Section 5.2 Your WeBWorK assignment for this section will ask you for right-hand Riemann sums, left-hand Riemann sums, and midpoint rule sums for a given value of n. This simply means that you should divide the interval [a, b] evenly into n subintervals, and use either right endpoints (xi [x i 1, x i ] defined by xi = x i ), left endpoints (xi [x i 1, x i ] defined by xi = x i 1 ), or the average of the two (xi [x i 1, x i ] defined by xi = x i +x i 1 2 ). The definite integral (if it exists) will return the same value regardless of the partition we choose, so frequently it is easiest to simply divide the interval [a, b] evenly and choose either right endpoints, left endpoints, or midpoints. Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Wed., 18 November 2015 11 / 13
Section 5.2 Theorem 5.2.1 If f is continuous on [a, b], or if f has only a finite number of jump discontinuities, then f is integrable on [a, b]; that is, the definite integral f (x) dx exists. b a Theorem 5.2.2 If f is integrable on [a, b], then b a f (x) dx = lim n f (x i ) x, i=1 where x = b a n and x i = a + i x. Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Wed., 18 November 2015 12 / 13
Section 5.2 Example Problems Evaluate the following definite integrals using Theorem 5.2.2, that is, find lim n n i=1 f (x i) x using right endpoints: 1 2 3 5 2 0 1 0 2 (4 2x) dx (x 2 + x) dx (x 3 3x 2 ) dx Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Wed., 18 November 2015 13 / 13