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Chapter 6: Integration Section 6.1: The Definite Integral Definition: If m and n are integers such that m ≤ n, then n X ak = am + am+1 + am+2 + · · · + an−1 + an . k=m The term k is called the index of summation, m is called the lower limit of the sum, and n is called the upper limit of the sum. Example: Write each sum in expanded form. (a) 6 X k=1 (b) 8 X 1 k+1 xk k=5 Example: Write each sum in sigma notation. √ √ √ √ √ (a) 3 + 4 + 5 + 6 + 7 (b) 1 − 1 1 1 1 1 + − + − 4 9 16 25 36 1 Theorem: (Summation Formulas) Suppose that c is a constant and n is a positive integer. Then 1. n X cak = c k=m 2. n X n X (ak ± bk ) = k=m 3. n X ak k=m n X ai ± k=m n X bk k=m c = cn k=1 4. n X k= k=1 5. n X n(n + 1) 2 k2 = k=1 n(n + 1)(2n + 1) 6 Example: Evaluate each sum. (a) 6 X k(k + 2) k=3 (b) 4 X 2k+1 k=1 2 (c) 100 X 4k k=1 (d) n X (5 − 2k) k=1 (e) n X (2k + 3)2 k=1 3 Area Under a Curve Suppose that f is a continuous function on the interval [a, b]. What is the area of the region S that lies under the curve y = f (x) from x = a to x = b? To approximate the area of this region, we begin by subdividing the interval [a, b] into n subintervals by choosing partition points x0 , x1 , x2 , . . . , xn such that a = x0 < x1 < x2 < · · · < xn−1 < xn = b. Then the n subintervals are [x0 , x1 ], [x1 , x2 ], . . . , [xn−1 , xn ]. This subdivision is called a partition of [a, b], and denoted by P . The length of the kth subinterval [xk−1 , xk ] is ∆xk = xk − xk−1 . The length of the longest subinterval is called the norm of P and denoted by ||P ||. That is, ||P || = max{∆x1 , ∆x2 , . . . , ∆xn }. Then we choose a representative point x∗k in each subinterval [xk−1 , xk ] and construct an approximating rectangle Rk with base ∆xk and height f (x∗k ). The area of the region S is approximated by the Riemann sum A≈ n X f (x∗k )∆xk . k=1 4 Example: Estimate the area under the graph of f (x) = 16 − x2 on [0, 4] using four equal subintervals and left endpoints. Is your estimate an underestimate or overestimate? 5 Example: Estimate the area under the graph of f (x) = 4 cos x on [0, π/2] using four equal subintervals and right endpoints. Is your estimate an underestimate or overestimate? 6 The Definite Integral To obtain a better approximation of the area of the region under the curve y = f (x) on the interval [a, b], we choose finer and finer partitions of the interval [a, b] and evaluate the Riemann sum. Definition: If f is a continuous function defined on a closed interval [a, b], let P be a partition of [a, b] with partition points x0 , x1 , x2 , . . . , xn , where a = x0 < x1 < x2 < · · · < xn = b. Choose representative points x∗k in [xk−1 , xk ] and let ∆xk = xk −xk−1 and ||P || = max{∆xk }. Then the definite integral of f from a to b is Z b f (x) dx = lim ||P ||→0 a n X f (x∗k )∆xk , k=1 provided that this limit exists. If the limit does exist, then f is called integrable on [a, b]. The function f (x) is called the integrand and a and b are called the limits of integration. Note: To evaluate a definite integral as the limit of a Riemann sum, it is often easier to use a regular partition. That is, a partition for which all the subintervals have the same length, b−a . n ∆x = If we choose x∗k to be the right endpoint, then x∗k = a + k∆x = a + b−a n k. Then the definite integral of f from a to b is Z b f (x) dx = lim a n→∞ 7 n X k=1 f (x∗k )∆x. Z Example: Evaluate 4 (x2 + 3x − 2) dx as the limit of a Riemann sum. 1 8 Note: If f (x) ≥ 0 on [a, b], then the definite integral of f from a to b represents the area under the graph of y = f (x) on [a, b]. However, if f (x) ≤ 0 on, then the definite integral of f from a to b is the negative of the area above y = f (x) on [a, b]. Example: Evaluate each definite integral by interpreting it in terms of signed areas. Z 2√ (a) 4 − x2 dx −2 Z 3 (2 − x) dx (b) −1 Z 3 |3x − 6| dx (c) 0 9 Theorem: (Properties of the Definite Integral) Suppose that f and g are integrable functions on [a, b] and c is a constant. Z b c dx = c(b − a) 1. a Z b Z b f (x) dx cf (x) dx = c 2. a a Z b b Z [f (x) ± g(x)] dx = 3. a Z b Z f (x) dx ± g(x) dx a a a f (x) dx = 0 4. a Z b a Z f (x) dx = − 5. f (x)dx a Z 6. b b c Z f (x) dx = b Z f (x) dx + a a f (x) dx c 7. If f (x) ≤ g(x) for a ≤ x ≤ b, then Z b Z f (x) dx ≤ a b g(x) dx. a 8. If m ≤ f (x) ≤ M for a ≤ x ≤ b, then b Z m(b − a) ≤ f (x) dx ≤ M (b − a). a Z Example: Evaluate 1 x2 cos x dx. 1 10 Example: Write each expression as a single definite integral. Z 7 Z 10 f (x) dx f (x) dx − (a) 2 2 Z 5 Z 0 f (x) dx − (b) −3 Z f (x) dx + −3 6 f (x) dx 5 Example: Find an upper and lower bound for the definite integral Z 2 √ x3 + 1 dx. 0 11