= N. Example 2: Calculate R 8,, and M for x =x on the interval 2,4. Which one is an overestimate? Underestimate?

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1 How would you find the area of the field with that road as a boundary? R R L R f ( = [ ] Eample : Calculate and for on the interval,. 6 Then calculate,, and M. Say a road runner runs for seconds at 5 m/s, then another second at 5 m/s, more seconds at m/s, and last more seconds at 5 m/s. How far does it travel total? Which one is an overestimate? Underestimate? L8 8 f ( [ ] Eample : Calculate R 8,, and M for = on the interval,. b a = N

2 Eample : Use linearity and formulas [] [5] to rewrite and evaluate the sums. a. 5 5 n b. j( j j= n= 5 Eample : Calculate the limit for the given function and interval. Verify your answer by using geometry. ( = [ ] a. limr f 9, N N

3 Eample : Calculate the limit for the given function and interval. Verify your answer by using geometry. b. limln f( = + [,] N Eample 5: Find a formula for R N and compute the area under the graph as a limit. a. f, ( = + [ ] Eample 5: Find a formula for R N and compute the area under the graph as a limit. ( = [ ] b. f,5 Calculus Notes 5.: The Definite Integral From 5.: When you add up the rectangles, do they all have to be the same width? For Riemann sum approimations, we rela the requirements that the rectangles have to have equal width.

4 Calculus Notes 5.: The Definite Integral Calculus Notes 5.: The Definite Integral Eample : Calculate R( f, P, C, where f( = 7 + cos on P : = < = < =.7< =.< = C: {.,.,.5,. } What is the norm P? [,] Calculus Notes 5.: The Definite Integral Calculus Notes 5.: The Definite Integral

5 Calculus Notes 5.: The Definite Integral Calculus Notes 5.: The Definite Integral Eample : Calculate 5 ( d and 5 d Calculus Notes 5.: The Definite Integral Calculus Notes 5.: The Definite Integral Eample : Calculate 7 ( + d 5

6 Calculus Notes 5.: The Fundamental Theorem of Calculus, Part I Recall from 5. Reading Eample 5: 7 d d + d = 7 7 d d = d d 7 7 Note that F ( = 7 = ( = 7 = ( 79 = 9 is an antiderivative of Calculus Notes 5.: The Definite Integral Eample : evaluate the integral using FTC I: 9 a. d ( + b. 5u u u du So we can re-write it as: 7 d F ( 7 F( = π c. sinθ dθ π Calculus Notes 5.: The Definite Integral Eample : Show that the area of the shaded parabolic arch in figure 8 is equal to four-thirds the area of the triangle shown. Calculus Notes 5.: The Fundamental Theorem of Calculus, Part II From 5. From 5. A A+B ( ( y = a b B 6

7 Calculus Notes 5.: The Fundamental Theorem of Calculus, Part II Proof of FTC II: Calculus Notes 5.: The Fundamental Theorem of Calculus, Part II Proof of FTC II continued: First, use the additive property of the definite integral to write the change in A( over [,+h]. + h A ( + h A( = f ( t dt f ( t dt + h = f ( t a a In other words, A(+h-A( is equal to the area of the thin section between the graph and the -ais from to +h. dt To simplify the rest of the proof, we assume that f( is increasing. Then, if h>, this thin section lies between the two rectangles of heights f( and f(+h. So we have: hf ( A( + h A( hf ( + h Area of small Area of section Area of big rectangle rectangle Now divide by h to squeeze the difference quotient between f( and f(+h A ( ( + h A( f f ( + h h We havelimf ( + h = f ( because f( is continuous, and limf ( = f ( h + h So the Squeeze Theorem gives us: A similar argument shows that for h<: Calculus Notes 5.: The Fundamental Theorem of Calculus, Part II Proof of FTC II continued: Calculus Notes 5.: The Fundamental Theorem of Calculus, Part II Equation Equation A ( A ( = f ( Equation and Equation show that eists and. 7

8 Calculus Notes 5.: The Fundamental Theorem of Calculus, Part II π Eample : Find F (, F (, and F, where F ( = tantdt Calculus Notes 5.: The Fundamental Theorem of Calculus, Part II Eample : Find formulas for the functions represented by the integrals. a. ( t 8t dt b. e t dt c. dt t Calculus Notes 5.: The Fundamental Theorem of Calculus, Part II Calculus Notes 5.: The Fundamental Theorem of Calculus, Part II Eample : Calculate the derivative. Eample : Eplain why the following statements are true. Assume f( is differentiable. a. d dθ ( cotu du b. d t dt d t + (a If c is an inflection point of A(, thenf ( c = (b A( is concave up if f( is increasing. (c A( is concave down if f( is decreasing. 8

9 6 6 ( t t 6 ( 6 6 ( ( ( v( t dt = t dt = = = 6 8+ = 8m 6 ( t dt = Position function: s( t Velocity function: s ( t = v( t Acceleration function: s t = v t = at ( ( ( Eample : Find the total displacement and total distance traveled in the interval [,6] by a particle moving in a straight line with velocity v ( t =t m/ s 6 v t dt=.5 6 tdt + t dt = = t m/s t = t dt.5 t dt v( t < fort < v( t > fort > =. =.5 6 ( ( 5. = t t 5. t t ( 8 ( 5. ( ( 5 Eample : The number of cars per hour passing an observation point along a highway is called the traffic flow rate q(t (in cars per hour. t a. Which quantity is represented by the integral q( t dt t t The integral q( t dt represents the total number of cars that passed t t, the observation point during the time interval [ ] t b. The flow rate is recorded at 5-minute intervals between 7: and 9: AM. Estimate the number of cars using the highway during this -hour period. Find the integral of the following: a. ( + 9 d b. cos( Find the integral of the following: d. e. cos( + d 9 d c. + d f. + d 9

10 Eample a: d du du = d d so so This equation is called the Change of Variables Formula. Eample b: d Calculus Notes 5.7 & 5.8: Further Transcendental Functions & Eponential Calculate dt t bdt = ln So = lnt = lnb lna= ln b b a a t a e d= e

11 Calculus Notes 5.7 & 5.8: Further Transcendental Functions & Eponential dt Eample : Evaluate the definite integral t+ Calculus Notes 5.7 & 5.8: Further Transcendental Functions & Eponential 5 d Eample : Evaluate the definite integral 5 5 Calculus Notes 5.7 & 5.8: Further Transcendental Functions & Eponential Eample : Evaluate the indefinite integral d + Calculus Notes 5.7 & 5.8: Further Transcendental Functions & Eponential Eample : Evaluate the definite integral d

12 Calculus Notes 5.7 & 5.8: Further Transcendental Functions & Eponential Calculus Notes 5.7 & 5.8: Further Transcendental Functions & Eponential Calculus Notes 5.7 & 5.8: Further Transcendental Functions & Eponential Eample 5: In the laboratory, the number of Escherichia coli bacteria grows eponentially with growth constant k=.( hours. Assume that bacteria are present at time t=. a. Find the formula for the number of bacteria P(t at time t. Calculus Notes 5.7 & 5.8: Further Transcendental Functions & Eponential Eample 6: Find all solutions of y = y. Which solution satisfies y = 9? ( b. How large is the population after 5 hours? Eample 7: The isotope radon- decays eponentially with a half-life of.85 days. How long will it take for 8% of the isotope to decay? c. When will the population reach,?

13 Calculus Notes 5.7 & 5.8: Further Transcendental Functions & Eponential Eample 8: Pharmacologists have shown that penicillin leaves a person s bloodstream at a rate proportional to the amount present. a. Epress this statement as a differential equation. b. Find the decay constant if 5 mg of penicillin remains in the bloodstream 7 hours after an initial injection of 5 mg. Calculus Notes 5.7 & 5.8: Further Transcendental Functions & Eponential Eample 9: A computer virus nicknamed the Sapphire Worm spread throughout the internet on January 5,. Studies suggest that during the first few minutes, the population of infected computer hosts increased eponentially with growth constant k=.85 a. What was the doubling time for this virus? b. If the virus began in computers, how many hosts were infected after minutes? minutes? s c. Under the hypothesis (b, at what time was mg of penicillin present? Eample : In 9, a remarkable gallery of prehistoric animal paintings was discovered in the Lascau cave in Dordogne, France. A charcoal sample from the cave walls had a C to C ratio equal to 5% of that found in the atmosphere. Approimately how old are the paintings? Calculus Notes 5.7 & 5.8: Further Transcendental Functions & Eponential Eample : Is it better to receive $ today or $ in years? Consider r=. and r=.7. Eample : Chief Operating Officer Ryan Martinez must decide whether to upgrade his company s computer system. The upgrade costs $, and will save $5, a year for each of the net years. Is this a good investment if r=7%?