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4.1: Antiderivatives and Indefinite Integration

Definition of an Antiderivative: A function F is an antiderivative of f on an interval I if [pic] for all x in I.

Ex: Consider [pic]. Can you think of a function, F, whose derivative is [pic]?

Possible antiderivatives are:

So, for any constant, C, _________________________ is an antiderivative of f. This is called the general antiderivative of f.

This means antiderivatives come in families/groups.

Theorem 4.1: Representation of Antiderivatives

If F is an antiderivative of f on an interval I, then G is an antiderivative of f on I if and only if

[pic] for all x in I, where C is a constant.

This C is called the constant of integration.

What is integration?

The process of finding an antiderivative is called antidifferentiation or indefinite integration.

Vocabulary: the following terms are synonymous

Antidifferentiation Indefinite Integration verbs/process

Antiderivative Indefinite Integral nouns/end result

Notation: [pic] is read “the antiderivative/integral of f with respect to x.”

Integration Rules:

Just remember: derivatives and integrals undo each other

Integration is the inverse of differentiation [pic]

Differentiation is the inverse of integration [pic]

See chart p. 244 for all of your rules—MEMORIZE the trig. ones

Power Rule: [pic]

**Very important that [pic], meaning we can’t yet find [pic]

Sum/Difference Rules: [pic]

Other things to remember:

Constants can be pulled in front/come along for the ride

NO product or quotient rule—evaluate integrals piece by piece with sum/difference rule

—re-writing the integrand with algebra to fit the rules is key

Examples: Find the indefinite integral.

1. [pic]

2. [pic]

3. [pic]

• Read Examples 2-6.

• The +C family effect: each different constant creates a _________________ shift of the base antiderivative graph (where C = 0). See page 247, Figures 4.2 and 4.3.

Differential Equations

Differential equation (in 3 forms): [pic] [pic] [pic]

To solve a differential equation you must find the original function whose derivative is given.

So, you’ll find a general antiderivative (with the +C).

General Solutions (1 equation, 3 forms):

Initial Conditions and Particular Solutions: If you are given a point (called an initial condition) on the graph of the solution function, you can find a particular solution to the differential equation.

Say our original problem (in 3 forms) is now this: Solve the following differential equations with given initial conditions.

[pic] [pic] [pic]

Particular Solutions (same problems from above):

[pic] [pic] [pic]

Vertical Motion: (derivation of formulas from 2.2 using differential equations)

Example 8: A ball is thrown upward with an initial velocity of 64 feet per second from an initial height of 80 feet. Find the position function giving height, s, as a function of time, t. [Remember -32 feet per second per second is acceleration due to gravity]

4.2: Area

Area: finding the area under a curve is our goal; we need summations to help us as you’ll see

In order to approximate area under a curve, one process is to inscribe and circumscribe rectangles, find the areas of those rectangles, and sum those up for an area approximation.

This process begins by subdividing the interval [a, b] into ‘n’ subintervals of equal width. So, the width of each rectangle is given by [pic].

An Upper Sum, S(n) A Lower Sum, s(n)

(overestimate) (underestimate)

A Right-hand Sum A Left-hand Sum

(heights are from right endpoints) (heights are from left endpoints)

**Right and left hand sums can be either upper or lower sums or neither.

**Don’t think they always have to go together as pictured—use the definitions in small print below the terms to decide what you have/ are finding.

It should be clear that [pic].

Theorem 4.3: Let f be continuous and nonnegative on the interval [a, b]. The limits as [pic] of both the lower and upper sums exist and are equal to each other; that is[pic].

Recall we had that [pic], meaning by Thm 4.3 and the Squeeze Thm, the actual area can be found by calculating either lower sums or upper sums. So you may choose to use right or left endpoints, whichever is easiest.

Now, to practice…

Find the lower and upper sums for the region bounded by the graph of [pic] and the x-axis on the interval [0, 2] using n = 4.

Midpoint Rule for approximations—uses the midpoint of each subinterval to generate/calculate the heights of the rectangles (instead of right or left endpoints)

Approximate the area of the region bounded by the graph of [pic] and the x-axis on the interval [0, 1] using the Midpoint Rule with n = 5.

4.3: Riemann Sums and Definite Integrals

For Riemann Sums, your rectangles or subintervals do NOT have to be of equal widths.

All sums found in 4.2 were Riemann sums, just cases of them where [pic] for all [pic].

Definition of a Definite Integral: If f is defined on the closed interval [a, b] and [pic] exists, then f is integrable (meaning you can take the integral) on [a, b] and the limit is denoted

[pic]

• [pic] is called the definite integral of f from a, the lower limit of integration, to b, the upper limit of integration.

Making Connections

Indefinite integral from 4.1 Definite integral from 4.3

[pic] [pic]

A family of functions A limit, a number

They are related and the Fundamental Theorem of Calculus in 4.4 will show us this.

Theorem 4.4: If a function f is continuous on [a, b], then f is integrable on [a, b].

This does NOT go the other way—[pic] is integrable on [-1, 1] but discontinuous at x = 0.

How are definite integrals related to area?

Ex: Evaluate the definite integral: [pic]____________. [Use your calculator.]

Now graph [pic] on [-3, 1], and find the area bounded by f and the x-axis.

Is the definite integral equal to your area calculation?

Theorem 4.5: The Definite Integral as the Area of a Region

If f is continuous and nonnegative on the closed interval [a, b], then the area of the region bounded by the graph of f, the x-axis, and the vertical lines x = a and x = b is given by

[pic].

Definition of Two Special Definite Integrals

• If f is defined at a, [pic] (think about area of region with zero width)

• If f is integrable on [a, b], then [pic] (need a minus to flip limits)

Theorem 4.6: Additive Interval Property

If f is integrable on three closed intervals determined by a, b, c, then [pic]

Theorem 4.7: Properties of Definite Integrals (inherited from summation properties)

If f and g are integrable on [a, b] and k is a constant, then the functions of kf and [pic] are integrable on [a, b], and

• [pic]

• [pic] (works for more than 2 functions)

Theorem 4.8: Preservation of Inequality (see Figure 4.26 and this makes perfect sense)

• If f is integrable and nonnegative on the closed interval [a, b], then

[pic].

• If f and g are integrable on the closed interval [a, b] and [pic] for every x in

[a, b], then

[pic].

Examples:

1. Evaluate the definite integral [pic]. (You may use geometry instead of the limit idea.)

**To re-familiarize yourself with area formulas of basic shapes, READ Example 3.

4.4: The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus formally states that differentiation and definite integration are inverses of each other. It shows how the antiderivative/indefinite integral we learned about in 4.1 is related to the definite integral from 4.3.

Read p. 275 for a good explanation of the inverse relationship between differential calculus and integral calculus through the tangent line problem and the area problem.

Theorem 4.9: The Fundamental Theorem of Calculus

If a function f is continuous on [a, b] and F is an antiderivative of f on the interval [a, b], then

[pic]

**READ Guidelines for Using the Fundamental Theorem of Calculus on p. 276. It explains nicely why we do not need a constant of integration when evaluating definite integrals.

Examples: Evaluate the following definite integrals.

1. [pic] 2. [pic]

Theorem 4.10: Mean Value Theorem for Integrals

If f is continuous on [a, b], then there exists a number c in [a, b] such that

[pic]

The f(c) value from the Mean Value Thm for Integrals is called the average value of f on [a, b].

So, to find this we must take [pic] and solve for f(c)…

[pic]

Definition of the Average Value of a Function on an Interval

If f is integrable on [a, b], then the average value of f on [a, b] is [pic].

If you want to see how this is really an average, read p. 279 from below the NOTE to Ex 4.

Examples:

3. Find the value of c in [1, 3] that causes [pic].

4. Find the average value of [pic]on [0, 3]. Does f actually take on this value at some x in the given interval? If so, state those x’s.

Recall this from 4.3 Notes:

Indefinite integral from 4.1 Definite integral from 4.3

[pic] [pic]

A family of functions A limit, a number

**If a and b aren’t constants, this can turn out to be a function…

Ex: Evaluate [pic].

Solution: [pic], a function of x, NOT just a #.

Now, find [pic].

Solution: [pic].

This is now generalized in what’s called the Second Fundamental Theorem of Calculus.

Theorem 4.11: The Second Fundamental Theorem of Calculus

If f is continuous of an open interval containing a, then, for every x in that interval,

[pic].

**If you have a more complicated version where u is a function of x, then it becomes

[pic] by Chain Rule

**You can even go a step further. If u and v are both functions of x, it becomes

[pic]

Examples:

5. Use the Second Fundamental Theorem to find [pic] for [pic].

6. Find [pic] for [pic].

7. Find [pic] for [pic].

The integral as an accumulation function—turn to p. 281 for discussion

This idea is a HUGE application of integration.

Ex: A train moves along a track at a steady pace of 75 mph from 7am to 9am. Express its total distance traveled as an integral and evaluate.

4.5: Integration by Substitution

Theorem 4.12: Antidifferentiation of a Composite Function

Let g be a function whose range is an interval I, and let f be a function that is continuous on I. If g is differentiable on its domain and F is an antiderivative of f on I, then

[pic].

If [pic], then [pic], and we have

[pic].

_____________________________________________________________________________

Why it makes sense…(substitutions are similar to chain rule in differentiation)

We can see that [pic].

Going backwards, [pic]

As you get better, you can recognize patterns and evaluate integrals without formally making substitutions. For more complicated integrands, u-substitution is used; this is called a change of variables.

Change of Variables: For u-substitution, you rewrite your integral in terms of u and du. To do this we use Thm 4.12 in the following way

[pic]

with proper u-sub.

Steps:

1. Choose u. (Usually, look to the most complicated part of the integrand and choose the inside of it for u.)

2. Find du.

3. Rewrite integral in terms of u and du only. (You might have to manipulate u and du further for this.)

4. Integrate

5. Rewrite answer in terms of original variable. (Use your u definition.)

Examples: Evaluate the following indefinite integrals.

1. (#14) [pic] 2. (#50) [pic]

3. (#62) [pic]

4. (#63) [pic]

Theorem 4.13: The General Power Rule for Integration

If g is a differentiable function of x, then [pic].

Equivalently, if [pic], then [pic].

Examples: Evaluate the following indefinite integrals; try to use pattern recognition and not formal u-substitution.

5. (#8) [pic] 6. [pic]

Theorem 4.14: Change of Variables for Definite Integrals

If the function [pic] has a continuous derivative on the closed interval [a, b], and f is continuous on the range of g, then

[pic]

You must change your limits of integration to u-limits when you substitute!!!

Examples: Evaluate the following definite integrals.

7. [pic] 8. [pic]

Theorem 4.15: Integration of Even and Odd Functions

Let f be integrable on [-a, a].

1. If f is an even function, then [pic].

**Recall a function is even if f(-x) = f(x) for all x in the domain of f.

2. If f is an odd function, then [pic].

**Recall a function is odd if f(-x) = -f(x) for all x in the domain of f.

This is a helpful time-saver if you remember the graphs of functions from previous classes.

Examples of that:

[pic] [pic] [pic]

**Just because we’ve now learned u-substitution does NOT mean you need it for every problem.

4.6: Numerical Integration

We’ve already used rectangles to approximate areas under curves and definite integrals. Here, we’re going to use trapezoids.

Any type of approximation of this sort is called numerical integration, and it’s needed when we can’t take an integral by other means.

The area of the trapezoid going from [pic] is [pic].

We’ll use trapezoids of equal widths because we’ve seen in 4.2-4.3 that as long as we take [pic] of our approximation, we’ll get an accurate result. So, [pic].

Theorem 4.16: The Trapezoidal Rule

Let f be continuous on [a, b]. The Trapezoidal Rule for approximating [pic] is given by

[pic]

**The coefficients on the f(x) pieces have the 1 2 2 2…2 2 1 pattern.

**Also, [pic].

Example: Use the Trapezoidal Rule to approximate [pic] with n = 4 and with n = 8.

Theorem 4.19: Error in the Trapezoidal Rule

If f has a continuous second derivative on [a, b], then the error, E, in approximating [pic] by the Trapezoidal Rule is

[pic].

**If you examine this you’ll see the error can be made negligible by letting [pic].

**For only a few trapezoids (small n), there can be significant error, depending on the function.

**READ Example 3.

Trapezoidal Rule vs. Midpoint Rule—these two tend to be the most accurate approximations without letting [pic].

Midpoint—you average the x’s at the endpoints of the subintervals and then take the function value at that average (midpoint) for the height

[pic]

Trapezoid—you average the function values at the endpoints of the subintervals

[pic]

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[pic]

[pic]

[pic]

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