Antiderivatives and Indefinite Integration Lesson 5.1

Reversing Differentiation An antiderivative of function f is a ________________F which satisfies __________ Consider the following: We note that two antiderivatives of the same function differ by a __________________

Reversing Differentiation General antiderivatives f(x) = 6x2 F(x) = 2x3 + C because ___________ = 6x2 k(x) = sec2(x) K(x) = ___________________ because K’(x) = k(x)

Differential Equation A differential equation in x and y involves x, y, and _____________________ of y Examples Solution – find a function whose ___________is the differential given

Differential Equation When Then one such function is The general solution is

Notation for Antiderivatives We are starting with Change to differential form Then the notation for antiderivatives is "The ______________of f with respect to x"

Basic Integration Rules Note the inverse nature of integration and differentiation Note basic rules, pg 286

Practice Try these

Finding a Particular Solution Given Find the specific equation from the family of antiderivatives, which contains the point (3,2) Hint: find the __________________, use the given point to find the value for C

Assignment A Lesson 5.1 A Page 291 Exercises 1 – 55 odd

Slope Fields Slope of a function f(x) Suppose we know f ‘(x) Example at a point a given by f ‘(a) Suppose we know f ‘(x) substitute different values for a draw short slope lines for successive values of y Example

Slope Fields For a large portion of the graph, when We can trace the line for a specific F(x) specifically when the C = -3

Finding an Antiderivative Using a Slope Field Given We can trace the version of the original F(x) which _______________________.

Vertical Motion Consider the fact that the acceleration due to gravity a(t) = -32 fps Then v(t) = -32t + v0 Also s(t) = -16t2 + v0t + s0 A balloon, rising vertically with velocity = 8 releases a sandbag at the instant it is 64 feet above the ground How long until the sandbag hits the ground What is its velocity when this happens? Why?

Rectilinear Motion A particle, initially at rest, moves along the x-axis at velocity of At time t = 0, its position is x = 3 Find the velocity and position functions for the particle Find all values of t for which the particle is at rest

Assignment B Lesson 5.1 B Page 292 Exercises 57 – 93, EOO

Area as the Limit of a Sum Lesson 5.2

Area under f(x) = ln x Consider the task to compute the area under a curve f(x) = ln x on interval [1,5] 1 2 3 4 5 x We estimate with 4 rectangles using the _________endpoints

Area under the Curve 1 2 3 4 5 x We can ________________our estimate by increasing the number of rectangles

Area under the Curve Increasing the number of rectangles to n This can be done on the calculator:

Generalizing a b In general … The actual area is where

Summation Notation We use summation notation Note the basic rules and formulas Examples pg. 295 Theorem 5.2 Formulas, pg 296

Use of Calculator Note again summation capability of calculator Syntax is:  (expression, variable, low, high)

Practice Summation Try these

Practice Summation For our general formula: let f(x) = 3 – 2x on [0,1]

Assignment Lesson 5.2 Page 303 Exercises 1 – 61 EOO (omit 45)

Riemann Sums and the Definite Integral Lesson 5.3

Review We partition the interval into n ____________ b We partition the interval into n ____________ Evaluate f(x) at _________endpoints of kth sub-interval for k = 1, 2, 3, … n f(x)

Review a b Sum We expect Sn to improve thus we define A, the ______________under the curve, to equal the above limit. f(x)

Riemann Sum Partition the interval [a,b] into n subintervals a = x0 < x1 … < xn-1< xn = b Call this partition P The kth subinterval is xk = xk-1 – xk Largest xk is called the _________, called ||P|| Choose an arbitrary value from each subinterval, call it _________

Riemann Sum Form the sum This is the Riemann sum associated with the function ______ the given partition ____ the chosen subinterval representatives ______ We will express a variety of quantities in terms of the Riemann sum

The Riemann Sum Calculated Consider the function 2x2 – 7x + 5 Use x = 0.1 Let the = left edge of each subinterval Note the sum

The Riemann Sum We have summed a series of boxes If the x were ____________________, we would have gotten a better approximation f(x) = 2x2 – 7x + 5

The Definite Integral The definite integral is the _______of the Riemann sum We say that f is _____________ when the number I can be approximated as accurate as needed by making ||P|| sufficiently small f must exist on [a,b] and the Riemann sum must exist

Example Try Use summation on calculator.

Example Note increased accuracy with __________ x

Limit of the Riemann Sum The definite integral is the ___________of the Riemann sum.

Properties of Definite Integral Integral of a sum = sum of integrals Factor out a _________________ Dominance

Properties of Definite Integral Subdivision rule f(x) a c b

Area As An Integral The area under the curve on the interval [a,b] A f(x) A a c

Distance As An Integral Given that v(t) = the velocity function with respect to time: Then _____________________ can be determined by a definite integral Think of a summation for many small time slices of distance

Assignment Section 5.3 Page 314 Problems: 3 – 49 odd

The Fundamental Theorems of Calculus Lesson 5.4

First Fundamental Theorem of Calculus Given f is _________________on interval [a, b] F is any function that satisfies F’(x) = f(x) Then

First Fundamental Theorem of Calculus The definite integral can be computed by finding an _________________F on interval [a,b] evaluating at limits a and b and _____________ Try

Area Under a Curve Consider Area =

Area Under a Curve Find the area under the following function on the interval [1, 4]

Second Fundamental Theorem of Calculus Often useful to think of the following form We can consider this to be a _______________ in terms of x View QuickTime Movie

Second Fundamental Theorem of Calculus Suppose we are given G(x) What is G’(x)?

Second Fundamental Theorem of Calculus Note that Then What about ? Since this is a _____________

Second Fundamental Theorem of Calculus Try this

Assignment Lesson 5.4 Page 327 Exercises 1 – 49 odd

Integration by Substitution Lesson 5.5

Substitution with Indefinite Integration This is the “backwards” version of the _____________________ Recall … Then …

Substitution with Indefinite Integration In general we look at the f(x) and “split” it into a ________________________ So that …

Substitution with Indefinite Integration Note the parts of the integral from our example

Example Try this … We have a problem … what is the g(u)? what is the du/dx? We have a problem … Where is the 4 which we need?

Example We can use one of the properties of integrals We will insert a factor of _____inside and a factor of ¼ __________to balance the result

Can You Tell? Which one needs substitution for integration? Go ahead and do the integration.

Try Another …

Assignment A Lesson 5.5 Page 340 Problems: 1 – 33 EOO 49 – 77 EOO

Change of Variables We completely rewrite the integral in terms of u and du Example: So u = _________ and du = _________ But we have an x in the integrand So we solve for x in terms of u

Change of Variables We end up with It remains to distribute the and proceed with the integration Do not forget to "_________________"

What About Definite Integrals Consider a variation of integral from previous slide One option is to change the limits u = __________ Then when t = 1, u = ___ when t = 2, u = ____ Resulting integral

What About Definite Integrals Also possible to "un-substitute" and use the ___________________ limits

Integration of Even & Odd Functions Recall that for an even function The function is symmetric about the ________ Thus An odd function has The function is symmetric about the orgin

Assignment B Lesson 5.5 Page 341 Problems: 87 - 109 EOO 117 – 132 EOO

Numerical Integration Lesson 5.6

Trapezoidal Rule Instead of calculating approximation rectangles we will use trapezoids More accuracy Area of a trapezoid a b • Which dimension is the h? Which is the b1 and the b2 b1 b2 h

Trapezoidal Rule Trapezoidal rule approximates the integral dx f(xi) f(xi-1) Trapezoidal rule approximates the integral Calculator function for f(x) ((2*f(a+k*(b-a)/n),k,1,n-1)+f(a)+f(b))*(b-a)/(n*2) trap(a,b,n)

Trapezoidal Rule Entering the trapezoidal rule into the calculator __________ must be defined for this to work

Trapezoidal Rule Try using the trapezoidal rule Check with integration

Simpson's Rule As before, we divide the interval into n parts n must be ___________ Instead of straight lines we draw _____________through each group of three consecutive points This approximates the original curve for finding definite integral – formula shown below a b •

Simpson's Rule Our calculator can do this for us also The function is more than a one liner We will use the program editor Choose APPS, 7:Program Editor 3:New Specify Function, name it simp

Simpson's Rule Enter the parameters a, b, and n between the parentheses Enter commands shown between Func and endFunc

Simpson's Rule Specify a function for ______________ When you call simp(a,b,n), Make sure n is an number Note the accuracy of the approximation

Assignment A Lesson 5.6 Page 350 Exercises 1 – 23 odd

Error Estimation Trapezoidal error for f on [a, b] Where M = _______________of |f ''(x)| on [a, b] Simpson's error for f on [a, b] Where K = max value of ___________ on [a, b]

Using Data Given table of data, use trapezoidal rule to determine area under the curve dx = ? x 2.00 2.10 2.20 2.30 2.40 2.50 2.60 y 4.32 4.57 5.14 5.78 6.84 6.62 6.51

Using Data Given table of data, use Simpson's rule to determine area under the curve x 2.00 2.10 2.20 2.30 2.40 2.50 2.60 y 4.32 4.57 5.14 5.78 6.84 6.62 6.51

Assignment B Lesson 5.6 Page 350 Exercises 27 – 39 odd 49, 51, 53

The Natural Log Function: Integration Lesson 5.7

Log Rule for Integration Because Then we know that And in general, when u is a differentiable function in x:

Try It Out Consider these . . .

Finding Area Given Determine the area under the curve on the interval [2, 4]

Using Long Division Before Integrating Use of the log rule is often in disguised form Do the division on this integrand and alter it's appearance

Using Long Division Before Integrating Calculator also can be used Now take the integral

Change of Variables Consider So we have Then u = x – 1 and du = dx But x = _________ and x – 2 = ______________ So we have Finish the integration

Integrals of Trig Functions Note the table of integrals, pg 357 Use these to do integrals involving trig functions

Assignment Assignment 5.7 Page 358 Exercises 1 – 37 odd 69, 71, 73

Inverse Trigonometric Functions: Integration Lesson 5.8

Review Recall derivatives of inverse trig functions

Integrals Using Same Relationships When given integral problems, look for these patterns

Identifying Patterns For each of the integrals below, which inverse trig function is involved?

If they are not, how are they integrated? Warning Many integrals look like the inverse trig forms Which of the following are of the inverse trig forms? If they are not, how are they integrated?

Try These Look for the pattern or how the expression can be manipulated into one of the patterns

Completing the Square Often a good strategy when quadratic functions are involved in the integration Remember … we seek _______________ Which might give us an integral resulting in the arctan function

Completing the Square Try these

Rewriting as Sum of Two Quotients The integral may not appear to fit basic integration formulas May be possible to ______________________into two portions, each more easily handled

Basic Integration Rules Note table of basic rules Page 364 Most of these should be committed to memory Note that to apply these, you must create the proper ________ to correspond to the u in the formula

Assignment Lesson 5.8 Page 366 Exercises 1 – 39 odd 63, 67

Hyperbolic Functions -- Lesson 5.9 Consider the following definitions Match the graphs with the definitions. Note the identities, pg. 371

Derivatives of Hyperbolic Functions Use definitions to determine the derivatives Note the pattern or interesting results

Integrals of Hyperbolic Functions This gives us antiderivatives (integrals) of these functions Note other derivatives, integrals, pg. 371

Integrals Involving Inverse Hyperbolic Functions

Try It! Note the definite integral What is the a, the u, the du? a = 3, u = _________, du = _______________

Application Find the area enclosed by x = -¼, x = ¼, y = 0, and Which pattern does this match? What is the a, the u, the du?

Assignment Lesson 5.9 Page 377 Exercises 1 – 29 EOO 37 – 53 EOO