Science One Integral Calculus. January 8, 2018

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1 Science One Integral Calculus January 8, 2018

2 Last time a definition of area Key ideas Divide region into n vertical strips Approximate each strip by a rectangle Sum area of rectangles Take limit for n

3 Last time The definition of the definite integral Consider the region under the curve y = f(x) above a, b. Take n vertical strips of equal width Δx = (b a)/n Consider n intervals [x 0, x 1 ],[x 1, x 2 ],[x 2,x 3 ], [x 89, x 8 ], [x ;9, x n ]. Build n rectangles and sum all areas S n = Δx f(x 1 *) + Δx f(x 2 *) +. + Δx f(x i *) + + Δx f(x n *) = ; 8? f(x 8 ) Δx where sample point x i * is any number in the interval x 89, x 8. Defn The area S of the region under the graph of f above [a, b] is ; S = lim f(x ; C 8? 8 ) Δx = f x dx Riemann Sum

4 The Riemann sum ; lim f(x ; C 8? 8 ) Δx = f x dx if x 8 = x 8 = a + i x right Riemann sum if x 8 = x 89 = a + (i 1) x left Riemann sum The Riemann sum method for computing an approximation of an integral

5 Interpret the following limit by first recognizing the sum as a Riemann sum for a function defined on [0,1] lim ; C ( + K + L + + ; ) ; ; ; ; ; Which of the following integrals is defined as the above limit? N a) x dx O c) O N P dx b) x dx O d) O P dx

6 U Consider sin 5x dx. Which of the following expressions represent O the integral as a limit of Riemann sums? ; U ; C ; a) lim 8? sin (π + WU 8 ; ) ; ; C 8? b) lim sin( U 8 ; ) ; U ; C 8? sin( U 8 ) ; ; c) lim ; U 8? sin( WU 8 ) ; C ; ; d) lim ; e) lim sin ( ; C 8? WU 8 ; )

7 Which of the following correctly expresses the limit ; K lim cos ; C 8? as a definite integral? ; ; K a) cos x dx O K b) cos x + 1 dx O L c) cos x dx K d) cos x dx K e) 2cos x dx

8 Terminology f x dx integrand

9 Terminology f x dx limits of integration

10 Terminology upper limit f x dx

11 Terminology f x dx lower limit

12 Terminology f x dx (dummy) integration variable

13 What if f (x) < 0 on a, b? If f (x) < 0 on [a, b] then curve is below x-axis... Riemann sum sum of the negatives of the areas of the rectangles that lie below the x-axis (and above the curve) f x dx < 0 signed area f x dx = area of region below x-axis, above curve y = f(x) on [a, b]

14 Using the definition of integral as area, compute K f x dx O = a) π b) π/2 c) Not defined d) -π e) -π/2

15 When does the limit of Riemann sums exist? ; lim a f(x ; C 8 ) 8? Δx = b f x dx Thrm Let f(x) be defined on [a, b]. If f(x) is continuous on [a, b], or f(x) has a finite number of jump discontinuities on [a, b] then the limit of Riemann sums exists and we say f(x) is integrable on a, b.

16 c Evaulate x 3 dx L = 3 x dx c + x 3 dx L = = 1 2 (2)(2) (1)(1) = 5 2

17 Some properties of the integral 1) [f x + g x ]dx 2) [f x g x ]dx = f x dx = f x dx 3) If f(x) = c = constant, then c dx Af x + Bg x dx = A f x dx + g x dx g x dx = c(b a) 1 f dx + B g x dx = b a

18 Some properties of the integral 1) [f x + g x ]dx 2) [f x g x ]dx = f x dx = f x dx + g x dx g x dx 3) If f(x) = c = constant, then c dx = c(b a) Af x + Bg x dx = A f x dx + B g x dx 4) f x dx 5) f x dx m 6) f x dx = f x dx = 0 + f x dx m = f x dx

19 Evaluating integrals using known areas.and the properties of the definite integral

20 Evaluating integrals using known areas c b f x dx = a) 4π b) 3π/4 c) 5π/4 d) 2π

21 3 + 1 x K dx 9 = 3dx x K dx 9 9 3dx = area rectangle of base 2 and height 3 = 2x3=6 9 1 x K dx = area half circle of radius 1 centred at origin = ½ π x K dx 9 = 6 + π/2

22 Comparison properties of the integral 1. If f(x) 0 on [a, b], then f x dx If f(x) g(x) on [a, b], then f x dx 3. If m f(x) M on [a, b], then m(b a) f x dx M (b a). g x dx. useful to bound integrals.

23 Estimate e 9Pz dx. O Observe e 9Pz is a decreasing function on [0, 1], that is e 9 e 9Pz 1 Thus We estimate e b e 9Pz dx O e 9Pz dx O

24 Recap What have we learned so far? The integral is defined as a limit of a Riemann sum Riemann sums can be constructed using any point in a subinterval Riemann sums provide a method to approximate an integral If f is (piece-wise) continuous on [a, b], f is integrable on [a, b] f x dx represents a signed area of the region Sad news Evaluating the limit of Riemann sums is hard! ood news there is an easier way to compute integrals

25 The undamental Theorem of Calculus Let f be continuous on an interval I containing a. P 1. Define x = f t dt on I. Then is differentiable on I with (x) = f(x). 2. Let be any antiderivative of f on I. Then for any b in I b f t dt = b (a)

26 TC part I The area function if f(x) is continuous on I let (x) = area under f(x) on [a, x] P (x) = f t dt (x) is called accumulation function

27 TC part I The area function Suppose f is continuous (and positive) on an interval containing a, then the area below the curve y = f(t) on [a, x] is P x = f t dt At what rate is (x) changing with respect to x? P 9 (P) (x) = lim O Observation (x + h) (x) is a difference of areas approximated by a rectangle of area h f f(x). In the limit h 0, we get (x) = f(x)..

Science One Integral Calculus. January 9, 2019

Science One Integral Calculus. January 9, 2019 Science One Integral Calculus January 9, 2019 Recap: What have we learned so far? The definite integral is defined as a limit of Riemann sums Riemann sums can be constructed using any point in a subinterval