5 Integration 5.1 Antiderivatives and Indefinite Integration Suppose that f(x) = 5x4. Can we find a function F (x) whose derivative is f(x)? Definition. A function F is an antiderivative of f on an interval I if F !(x)=f(x) for all x in I. 5 5 5 Note. Observe that F1(x)=x , F2(x)=x + 2, and F3(x)=x √7 are all antiderivatives of f. To represent all possible antiderivatives of f(x), we write F (x)=−x5 + C where C is a constant. Theorem 5.1. If F is an antiderivative of f on an interval I, then G is an antiderivative of f on the interval I if and only if G is of the form G(x)=F (x)+C, for all x in I where C is a constant. Note. We call C the constant of integration and G the general antiderivative of f. Notation. From the last section, for dy = f(x) dx we wrote dy = f(x)dx The process of finding all solutions to this equation is called indefinite integration or antidif- ferentiation. We write y = 1 5 Integration Note. There is an “inverse” relationship between differentiation and indefinite integration. That is F !(x)dx = • ! If f(x)dx = F (x)+C, then • ! d f(x)dx = dx "! # Basic Integration Rules Compare these with the basic differentiation rules that we already know. 1. 0dx = ! 2. k dx = ! 3. kf(x)dx = ! 4. f(x) g(x) dx = ± ! $ % 5. xn dx = ! 1 6. dx = x ! 7. cos x dx = ! 8. sin x dx = ! 9. sec2 x dx = ! 10. csc2 x dx = ! 11. sec x tan x dx = ! 12. csc x cot x dx = ! 2 5.1 Antiderivatives and Indefinite Integration Example 1. Evaluate 5dx Example 2. Evaluate 5x dx ! ! 1 Example 3. Evaluate dx Example 4. Evaluate √3 x dx x4 ! ! 5 Example 5. Evaluate 2 cos x dx Example 6. Evaluate dx x ! ! 3 5 Integration Example 7. Evaluate dx Example 8. Evaluate (x + 4) dx ! ! Example 9. Evaluate (x2 +5x + 4) dx ! 3x +2 sin x Example 10. Evaluate dx Example 11. Evaluate dx √x cos2 x ! ! 4 5.1 Antiderivatives and Indefinite Integration Example 12. Near the surface of the earth, the acceleration of a falling body due to gravity is 32 feet per second per second, provided that air resistance is neglected. If an object is thrown upward− from an initial height of 1000 feet with an initial velocity of 50 feet per second, find its velocity and height four seconds later. 5 5 Integration 5.2 Area Sigma Notation The sum of n terms a1, a2, a3, . , an may be denoted by n ai = i=1 & th where i is the index of summation, ai is the i term (or general term) of the sum, and 1 and n are the lower and upper bounds of summation respectively. 5 Example 1. Expand i. i=1 & 6 Example 2. Expand i2. i=3 & n 1 Example 3. Expand (k2 1). n − &k=1 n Example 4. Expand f(xi)∆x. i=1 & 6 5.2 Area Properties of Sigma Notation n 1. kai = i=1 & n 2. (a b )= i ± i i=1 & Theorem 5.2. n n 1. c = 2. i = i=1 i=1 &n &n 3. i2 = 4. i3 = i=1 i=1 & & n i2 +1 Example 5. Find the summation formula for . Use your result to evaluate the sum for n3 i=1 n = 10, 100, 1000, 10000. & 7 5 Integration Example 6. Use four rectangles to find two approximations of the area of the region lying between the graph of f(x)=2x and the x-axis between the the vertical lines x = 0 and x = 2. y y 4 4 3 3 2 2 1 1 x x 12 12 8 5.2 Area To approximate the area under the curve of a nonnegative function bounded by the x-axis and the line x = a and x = b... y x y x y x 9 5 Integration y x y x y x 10 5.2 Area Theorem 5.3. Let f be continuous and nonnegative on the interval [a, b]. The limits of both the lower and upper sums exist and lim s(n)= n →∞ where ∆x =(b a)/n and f(m ) and f(M ) are the minimum and maximum values of f on the − i i subinterval. Definition. Let f be continuous and nonnegative on the interval [a, b]. The area of the region bounded by the graph of f, the x-axis, and the vertical lines x = a and x = b is area = where ∆x =(b a)/n. − y x Example 7. Find the area of the region bounded by the graph of f(x)=x2, the x axis, and the − vertical lines x = 0 and x = 1. 11 5 Integration 5.3 The Riemann Sum and the Definite Integral The following definition allows us to find the area under a function when the width of the rectangles are not all the same. Definition. Let f be defined on the closed interval [a, b], and let ∆be a partition of [a, b] given by a = th th where ∆xi is the width of the i subinterval. If ci is any point in the i subinterval, then the sum n f(ci)∆xi i=1 & is called a Riemann sum of f for the partition ∆. Note. This is a generalization of the upper and lower sum method we developed earlier. We define the width of the largest subinterval of a partition ∆to be the norm of the partition denoted by ∆ . # # If every subinterval is of equal width, then the partition is said to be regular and the norm is given by ∆ = # # 12 5.3 The Riemann Sum and the Definite Integral If we are using a general partition, then the norm of the general partition is related to the number of subintervals of [a, b] by b a − ∆ # # and as ∆ 0, then this implies that n . # #→ →∞ The converse of this statement is not true in general. It is only true in the case of a regular n partition. For instance, consider the partition ∆n =2− of the interval [0, 1]. Thus, n lim f(ci)∆xi = L ∆ 0 % %→ i=1 & exists provided that for each ε> 0, there exists a δ> 0 with the property that for every partition with ∆ <δ implies that # # n L f(ci)∆xi <ε ' − ' ' i=1 ' ' & ' for any c in the ith subinterval of ∆. ' ' i ' ' Definition. If f is defined on the closed interval [a, b] and the limit n lim f(ci)∆xi ∆ 0 % %→ i=1 & exists (as described above), then f is integrable on [a, b] and the limit is denoted by n lim f(ci)∆xi = ∆ 0 % %→ i=1 & and is called the definite integral of f from x = a to x = b. The number a is the lower limit of integration and the number b is the upper limit of integration. 13 5 Integration Theorem 5.4. If a function f is continuous on the closed interval [a, b], then f is integrable on [a, b]. 2 Example 1. Evaluate 2x dx 1 !− 14 5.3 The Riemann Sum and the Definite Integral Theorem 5.5. If a function f is continuous and nonnegative on the closed interval [a, b], then the area of the region bounded by the graph of f, the x-axis, and the vertical lines x = a and x = b is given by area = 5 Example 2. Sketch the region corresponding to 3dx and evaluate the integral by using the 1 geometry of the region. ! y 5 4 3 2 1 x 12345 3 Example 3. Sketch the region corresponding to √9 x2 dx and evaluate the integral by using 3 − the geometry of the region. !− y 5 4 3 2 1 x 5 4 3 2 1 12345 − − − − − 15 5 Integration Theorem 5.6. If a function f is integrable on the three closed intervals determined by a < c < b, then b f(x)dx = !a y x 4 Example 4. Evaluate x dx. 4 | | !− y 5 4 3 2 1 x 5 4 3 2 1 12345 − − − − − Theorem 5.7. Suppose that k is a real number and f and g are integrable on [a, b]. Then, the function kf and f g are integrable on [a, b] and ± b 1. kf(x)dx = !a b 2. f(x) g(x) dx = ± !a $ % Example 5. For the definite integral, we can write 3 (x2 +5x + 6) dx = !1 16 5.3 The Riemann Sum and the Definite Integral Theorem 5.8. 1. If f is integrable and nonnegative on the closed interval [a, b], then 2. If f and g are integrable on the closed interval [a, b] and f(x) g(x) for every x in [a, b], then ≤ y x Properties of the Definite Integral 1. If f is defined at x = a, then 2. If f is integrable on [a, b], then Example 6. For the given definite integrals, we can write π ex dx = !π and 0 (x + 4) dx = !3 17 5 Integration 5.4 The Fundamental Theorem of Calculus Theorem 5.9 (Fundamental Theorem of Calculus Part I). If a function f is continuous on the closed interval [a, b] and F is an antiderivative of f on [a, b], then b f(x)dx = !a Note. We lose the constant of integration when performing definite integration. 18 5.4 The Fundamental Theorem of Calculus 2 Example 1.