The Definite Integral. Day 5 The Fundamental Theorem of Calculus (Evaluative Part)

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1 The Definite Integral Day 5 The Fundamental Theorem of Calculus (Evaluative Part)

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4 Practice with Properties of Integrals 5 Given f d 5 f d f d 0 f d f d f d

5 F 3 t dt Evaluate F 3 0 F t dt When the upper and lower bounds are the same, d 0

6 Given F f t dt f t dt 7 f t dt f t dt. F 3. F 5 3. F 0. F 8 F 5

7 Given F f t dt f t dt 7 f t dt f t dt 3. F 3 f t dt F 5 f t dt F 0 f t dt F 8 F 5 f t dt f t dt

8 The Definite Integral Day 5 Introduction to the Fundamental Theorem of Calculus (Evaluative Part)

9 Today s Learning Outcomes Define and find antiderivatives Make connections between derivatives, antiderivatives and integrals State and apply FTC in order to evaluate definite integrals Evaluate definite integrals without using fnint or Geometry

10 Terminology: A function F() is an antiderivative of a function f() if F ()=f().

11 For each problem, write 3 functions whose derivatives would be the derivative given. ) y' = 3 + ) y' = cos 3) y' = e ) y' = BE CREATIVE!!!!

12 For each problem, write 3 functions whose derivatives would be the derivative given. ) y' = 3 + y = c ) y' = cos y = sin + c 3) y' = e y = e + c ) y' = y = c These are known as antiderivatives

13 Find antiderivatives for f g 3 h sin An antiderivative is a function which would have the given function as its derivative.

14 Solutions f 9 F 3 C F ()=f() g 3 G C G ()=g() h sin H cos C H ()=h() C is a constant

15 Using mathematical symbols... Finding the antiderivative of a function is also called integration F( ) C f ( ) d NOTE: There are no boundaries on this integration. This is referred to as an indefinite integral. MORE later.

16 What have we learned so far about integration? Finding the area between a curve and the -ais. Estimating integrals using Riemann Sums Finding eact values of integrals using the calculator-- fnint

17 What have we not learned, YET? How to find eact integral values without using fnint when the region bounded by the function and the -ais is not some shape that we can apply geometry area formulas to.

18 The derivative of position is velocity The antiderivative of velocity is position

19 Let vt velocity 0 end start distance traveled v t dt distance traveled end position - start position end start v t dt end position - start position From earlier, position is the antiderivative of velocity. end start v t dt V end V start where V is an antiderivative of v.

20 Fundamental Theorem of Calculus (FTC) Mathematicians had known about derivatives and how to use them for hundreds of years. It was also known that integrals and areas were the keys to solving many problems, but no one knew how to solve them. Until..FTC The significance of the FTC is that it unites Differential and Integral Calculus and, in the process, tells us how to evaluate integrals.

21 Fundamental Theorem of Calculus, (Evaluation Part) If f is continuous at every point of a, b, and if F is any antiderivative of f on a, b, then b a f d F b F a MUST meet these conditions!!!!!

22 Again, for emphasis Antiderivatives can be used to evaluate integrals. ò b a f ( )d = F b ( ) - F( a) Where F is an antiderivative of f.

23 BEAUTIFUL! Now we have the power to evaluate statements like ò 0 ( 3-9 ) d without using fnint or Geometry Lets Try: 3 First step---figure out the antiderivative, F(), of the integrand, f() Net find F(b) and F(a). Finally, find F(b)-F(a)

24 Formatting our work: d ( 3 ) 0 3(0 3 ) = -0

25 We used the reverse power rule to find the antiderivative b a c d c b a b n c d c n b a n a

26 Practice 3 d 5 7 d

27 Thumbs up? Checkpoint for Understanding Define and find antiderivatives Make connections between derivatives, antiderivatives and integrals State and apply FTC in order to evaluate definite integrals Evaluate definite integrals without using fnint or Geometry

28 MORE Learning Outcomes Find antiderivatives of trig functions Find antiderivatives of absolute value functions

29 Finding Trig antiderivatives b a b a b a E: sin d sec cos d b d csc csc cot d sec tan d z a 5 sin d b a a b d

30 You Try: cos d 0 sec d

31 What about Absolute Value Problems? ò - d Discuss with your partner what might need to be done!!!

32 Remember an absolute value function is really a piecewise function.,, ( ) = ò - - d + - SO.. ò - d ò ( ) d

33 Other Eamples Remember your Algebra Skills 8 d 8 d d d 3 3

34 Integrate using the power rule 5 d 3

35 Integrate using the power rule???? 5 d

36 You can t use the power rule 5 0 d 0 5

37 What function has a derivative of f( )? 5 d ln 5 ln(5) ln() ln5

38 d ln But we cannot evalute the natural logarithm of negative numbers. So, a simple solution would be: d ln

39 Understanding why we need absolute value.. d ln c f( ) What is the domain of? F( ) ln What is the domain of? (-,0) È( 0, ) ( 0, ) f( ) is symmetrical about the y-ais and integrable. Using the absolute value allows us to do this.

40 d ln d is the area below the -ais and above the curve f( ) Due to the symmetry of f()=/ this area should be equal to d

41 Proof that: d ln d -ò d = - ln = - ln - ln [ ] [ ] - ln - = ln - - ln - = ln - ln = - ln = - ln = ln = ln = ln( ) - = - ln

42 Conclusion b a d ln b a

43 Others k d k ln C e d e C a d a ln a C

44 You Try! 5 d d ln 5 5 ln 0 6 e d ln e 0 6