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Section I The Basics

Page 3Antiderivatives

If I ask you what is the derivative of x, you will say 1.

If I ask you what is the antiderivative of x, and if you think I am asking you for the function such that the derivative of that function is x, then you are right. Now the derivative of what function is x? You might first guess it’s x2. However, the derivative of x2 is 2x, not x. So you might change x2 (just a bit) to ____ so that the derivative can be x.

Now, what are the antiderivative of the following functions?

1. Antiderivative of 1 = _____ 2. Antiderivative of x2 = _____

3. Antiderivative of 1 + x2 = _____ 4. Antiderivative of 4x3 = _____

5. Antiderivative of 3x4 = _____ 6. Antiderivative of xn = _____

7. Antiderivative of kxn = _____ 8. Antiderivative of cos x = _____

9. Antiderivative of sin x = _____ 10. Antiderivative of sin x – cos x = _____

Page 4

Definition of the f(x) dx: We write

and say the integral of f(x) is F(x) if the derivative of F(x) is f(x). F(x) is also called the antiderivative of f(x).

Integrals—Notation and Definition

For example, If you are asked for 2x dx? Ask yourself the derivative of what function yields 2x, and obviously, the function is x2 since the derivative of x2 is 2x. Therefore, 2x dx = x2.

But wait… the derivative of functions such as x2 + 1, x2 + 2006 and x2 is also 2x. So what is the 2x dx? As we can see, the antiderivative of 2x is not unique. Nevertheless, each of these functions contains ___ and they all differ by a _________. Therefore, we say 2x dx is _______ where C is a constant.

)()( xFdxxf

2

1

xx

Recall:

Note:

1. ( )

2. cos

3. sin

4. ( ) ( )

5. ( )

nx dx n

xdx

xdx

f x g x dx

k f x dx

More examples:

1. 3x – 2 dx =

2. dx =

3. cos x – sin x dx =

4. sec 2 x + sec x tan x dx =

6. dx

Page 5Integrals—Definite vs. Indefinite

There are mainly two types of integrals: definite integrals and indefinite integrals. Any other types are just subtypes of these two.

What we did earlier is called indefinite integrals, which are also shown below on the left. The definite integrals are shown on the right. See if you can spot any difference between the two types?

Indefinite Integrals vs. Definite Integrals?

1. x2 dx

2. cos x dx

3. 3t2 – 2t + 1 dt

1

0

2.1 dxx

2

6

cos.2

xdx

3

1

2 123.3 dttt

Terminology:

is read as the (definite) integral of f from a to b, and in this notation,

1. is the integral sign (introduced by Leibniz, one of two cofounders of calculus; the other cofounder was _________)

2. The a and b are the limits of integration, where a is the ______ limit and b is the ______ limit.

3. f(x) is called the integrand.4. The symbol, dx, only signifies the integration is with respect to x.

b

adxxf )(

Page 6How to Evaluate Definite Integrals

Recall that if I ask you x2 dx? You say ________. Now how do we evaluate ?

This is how: Example:

More examples:

1

0

2.1 dxx

2

6

cos.2

xdx

3

1

2 123.3 dttt

1. Find the antiderivative function for the definite integral the same ways as if it’s an indefinite integral, only you don’t need to write the “+ C”.

2. Plug the upper limit into the antiderivative function, then do the same for the lower limit.

3. Subtract the two numbers from step 2, that will the answer to the definite integral.

1 20

x dx

1 20

x dx

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Section II Finding the Area of a Region

Page 8The Area of a Region Bounded by Straight-Line Curves

Example 1: Find the area of the region bounded by the x-axis, the y-axis, and the lines y = 4 and x = 2.

Example 2: Find the area of the region bounded by the x-axis, and the lines y = 2x and x = 2.

Example 3: Find the area of the region bounded by the x-axis and the lines y = 2x, x = 1 and x = 2.

Page 9The Area of a Region Bounded by Curves (At Least One is not Straight)

Example: Find the area bounded by the x-axis, the curve y = x2 and the line x = 1.

12

1x

14

12

34

1y

14

12

34

1x

14

12

34

1y

18

14

38

12

58

34

78

1x

14

12

34

1y

Page 10How to Improve the Approximation of the Area?

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

n = __ n = 32 n = 100

As we can see, even though the sum of the areas of the rectangles is always an ___________ of the area of the region, as the number of the rectangle increases, the sum of the areas of the rectangles is getting closer to the area of the region. The following table shows the sum of the area of the rectangles when n, the number of rectangles, is 2, 4, 8, 16, 100, 1,000, and 10,000. See if you can predict of the exact area of the region is?

n Area* 2 5/8 or 0.625 4 15/32 or 0.46875 8 51/128 or 0.398438 16 187/512 or 0.365234 100 0.33835 1,000 0.333834 10,000 0.333383

* Note: The calculation for the area column is made possible by Mathematica.

From the table, we can predict the exact area of the region is ______or ___.

Example: Find the area bounded by the x-axis, the curve y = x2 and the line x = 1.

Page 11How Many Rectangles Do We Need?

Example: Find the area bounded by the x-axis, the curve y = x2 and the line x = 1.As we can see, the more rectangles we use, the better the approximation for the area of the region. It is easy and feasible to calculate the total area of a couple of rectangles, however, it will be tedious and time consuming to calculate manually the area of a lot of rectangles (say 100, 1,000, 10,000, etc). Furthermore, to find the exact area of region, we must use ___________ many rectangles. How can we find the area of infinitely many rectangles?

This is how: 1. We are dividing the region into n subdivisions, for each subdivision, we draw a rectangle. (When you

draw the picture out, despite n can be any positive integer, choose n to be small, say 4 or 5).2. For each rectangle, we ask ourselves what the base and the height of the rectangle are. Notice that

from one rectangle to next, while the base remains the same (it will be __ in this case, and in general, it will be _____), meanwhile, the ______ is changing because a non-constant function will have different heights at different x-values .

3. Find the sum of the areas of the rectangles—it will be an expression in terms of n, and simplify this expression.

4. In order to find the exact area of the region, take the limit as n approaches _________, because we need infinitely many rectangles to get the exact area of the region!

Page 12Gosh, Is There a Short Way?

Example: Find the area bounded by the x-axis, the y-axis, the curve y = x2 and the line x = 1.Yes, to find the exact area of region, just evaluate the definite integral of __ from __ to __, i.e.,

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

And in general, If the function f(x) is continuous and nonnegative on a closed interval [a, b], the exact area of region bounded by the x-axis, y = f(x), x = a and x = b is:

More Examples: 1. Find the area bounded by the x-axis, the curve y = x3 and the line x = 2.

2. Find the area bounded by the x-axis, the curve , from x = 1 to x = 4.

3. Find the area enclosed by the x-axis, f(x) = sin x from x = 0 to x = .

2

1

xxy

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Section III Different Ways to Approximate the Area of a Region

Page 14

14

12

34

1x

14

12

34

1y

Right Endpoint Method vs. Left Endpoint Method

In Section II, we learn to how approximate the area of a region (bounded by the x-axis, the curve y = x2 and the line x = 1) by using circumscribed rectangles. Hence, in some textbooks, this method is called Circumscribed Rectangles Method. However, in our textbook, it is called the Right Endpoint Method since the height of each rectangle is determined by the right endpoint after we subdivide the region into n intervals (see below for depiction when n = 4).

14

12

34

1x

14

12

34

1y

14

12

34

1x

14

12

34

1y

Of course, if there is a Right Endpoint Method, there is a ____________ Method, in which case, the height of each rectangle will be determined by the __________ after we subdivide the region into n intervals (once again, let n = 4). Let’s find the approximate area:

14

12

34

1x

14

12

34

1y

14

12

34

1x

14

12

34

1y

14

12

34

1x

14

12

34

1y

Page 15Right Endpoint Method vs. Left Endpoint Method (cont’d)

Recall that the exact area of the region (bounded by the x-axis, the y-axis, the curve y = x2 and the line x = 1) is __. For n = 4 (i.e., 4 rectangles), the Right Endpoint Method (REM) yields an estimate of ___, which is an __________ of the exact area, whereas the Left Endpoint Method (LEM) yields an estimate of _____, which is an __________ of the exact area. However, to say REM is always an overestimate and LEM is always an underestimate is not correct, because it also depends on the graph of the function (or curve). For example, if the region is bounded by the x-axis, y = 1/x, and the lines x = 1 and x = 5, the REM will be an __________ of the exact area and LEM will be an ___________ of the exact area.

1 2 3 4 5 6

0.20.40.60.81

1.21.4

1 2 3 4 5 6

0.20.40.60.81

1.21.4

Furthermore, if the function is both increasing and decreasing on a closed interval, then we can’t tell whether the estimate by REM (or by LEM) is an overestimate or an underestimate. For example, the region is bounded by the x-axis, y = sin x from 0 to :

4

2

3 4

0.2

0.4

0.6

0.8

1

4

2

3 4

0.2

0.4

0.6

0.8

1

REM: LEM:

REM: LEM:

Page 16Other Methods: Midpoint Rule and “Average” Method

In the previous slides, we subdivide the region into sections and for each section we either take the left endpoint or the right endpoint to construct our rectangles. However, in general, it doesn’t matter—so long as, for a section, we pick a point that is on the curve. See diagram on the right (top). Such a point is called a sample point. It’s only because in math we want to do things systematically, that’s why we choose the left endpoints, or the right endpoints or the __________ as our sample points. See diagram on the right (below). When we construct the rectangles by sampling the midpoints, the method is called Midpoint Rule (MR).

Example: Use MR and n = 4 to approximate the area of the bounded by the x-axis, the y-axis, the curve y = x2 and the line x = 1.

14

12

34

1x

14

12

34

1y sample points

14

12

34

1x

14

12

34

1y

Recall the exact area of the region is 1/3, so the Midpoint Rule (compared to LEM and REM) is by far the most accurate. However, if we have done the LEM and REM, (in this case, LEM = 7/32 and REM = 15/32), there is something we can do to these two numbers (without increasing n, the number of rectangles) to make the approximation more accurate. What is it?

Page 17Can We Just Integrate—Why Do We Need These Approximation Rules?

Recall that we can find the area of the region bounded by the x-axis, the curve y = x2 and the line x = 1, we just do:

Why do we need to use REM (Right-Endpoint Method), LEM (Left-Endpoint Method) or MR (Midpoint Rule)? Each of these method is only an approximation of the area of the region, not the exact area (unless we use infinitely many rectangles).

The reason is: Not every function can be easily integrated as x2. Some functions are difficult to integrate while some functions CAN’T be integrated. For example,

1. Find the area of the region bounded by the x-axis and y = x2 sin x from x = 0 to .

2. Find the area of the region bounded by the x-axis and y = esin x from x = 0 to 4.

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

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Section IV Area vs. Definite Integral

Page 19Does Definite Integral Always Yield the Area Under the Function?

Example: Evaluate the integral .

So the area under the function y = sin (2x) from 0 to is __? Yes! In a way, it is... Let’s look at the graph:

Example: Find the area of the region bounded by the function y = sin (2x) and the x-axis from 0 to .

Solution: If the area of any 2-dimensional region must be positive, simply using the definite integral will not give us the area of the region. Since the answer to the definite integral is 0 and we know the area can’t be 0! What should we do then?

We still going to use definite integral to find its area, but we are not going to find it use one definite integral, instead we are going to use two of them (hint: divide the region into two subregions).

0)2sin( dxx

0)2sin( dxx

Page 20Finding Area Bounded by a Function and the x-Axis

Once again, to find the area bounded by a continuous function f(x) and the x-axis from a to b:

i. If f(x) is positive (or nonnegative) on [a, b], then the area A = .

ii. If f(x) is negative (or nonpositive) on [a, b], then the area A = or A = .

iii. If f(x) is both positive and negative on [a, b], then (see the following):

b

adxxf )(

b

adxxf )(

b

adxxf )(

a bc a bc1 c2 c3

A = or

A=

A = or

A=

Therefore, if the problem asks for the evaluation of the definite integral of a function f(x) from a to b, there is no need to consider the graph of f—we just integrate and plug in the a and b . However, if the problem asks for the area bounded by f and the x-axis from a to b, not only we need to know the graph of f from a to b, but also we need to know the __________ of f between a and b. (In our textbook, the definite integral is the “net” area.)

Example: 1. Evaluate:

2. Find the area bounded by x2 – 5x + 6 and the x-axis from x = 1 to x = 4.

4

1

2 65 dxxx

b

adxxf )(

Page 21Finding Area Bounded by Two Functions

If we have two continuous functions f(x) and g(x) with the property that f(x) g(x) on a closed interval [a, b], then the area bounded by these two functions from a to b is

Example: Find the area bounded by f(x) = x2 and g(x) = x on [0, 1].

b

A =

a

y = f(x)

y = g(x)

However, in general, if we have two continuous functions f(x) and g(x) on a closed interval [a, b], then the area bounded by these two functions from a to b (see below)

Besides knowing the graph of the two functions, we also need to know their ____________ _______. To find it/them, we simply set the two functions equal to each other and solve for x.

Example: Find the area bounded by f(x) = x3 and g(x) = x on [–1, 1].

b

A =

a

y = g(x)y = f(x)

Page 22Finding Area Bounded by Two Curves

There’s nothing in the book that says to find the area of a region bounded by two curves mans to integrate from x = a to b (i.e. , horizontally). We can integrate from y = c to d (i.e., vertically) if we want to. Furthermore, we need to integrate a region vertically if it is easier this way than integrating it horizontally.

A = c

x = f(y)x = g(y)

d

Example 2: Find the area bounded by y = x, , y = 1 and the x-axis. 2 1 1y x

Example 1: Find the area bounded by y = x and .2 1 1y x

Example 3: Find the area bounded by x = ½y2 – 3 and x = y + 1.

Page 23

Section V Volumes by Revolution

Page 24

Disk Method:If f(x) is continuous and nonnegative on [a, b], then the volume generated by revolving the region under f from a to b is:

Volume by Revolving a Region under a Function (and Above the x-Axis)

Example 1: Find the volume of a solid generated by revolving the region under f(x) = 3 from x = 0 to x = 4.

Example 2: Find the volume of a solid generated by revolving the region under f(x) = x from x = 0 to x = 4.

Example 3: Find the volume of a solid generated by revolving the region under f(x) = x2 from x = 0 to x = 2.

b

a

b

a

dxxfV

or

dxxfV

2

2

)]([

)]([

where f(x) is the ______ of a cross-section and [f(x)]2 is the ______ of the cross-section.

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Washer Method:If f(x) and g(x) are continuous and nonnegative on [a, b] with f(x) g(x) on [a, b], then the volume generated by revolving the region between f and g from a to b is:

Volume by Revolving a Region Between Two Functions

Example 1: Find the volume of a solid generated by revolving the region between f(x) = x2 and g(x) = x from x = 0 to x = 1.

Example 2: Find the volume of a solid generated by revolving the region between f(x) = x2 + 1 and g(x) = x from x = 1 to x = 4.

xExample 3: Find the volume of a solid generated by revolving the region under f(x) = x and g(x) = from x = 1 to x = 4.

b

a

b

a

dxxgxfV

or

dxxgxfV

22

22

)]([)]([

))]([)](([

Page 26

Section VI Integration of Special Functions

Page 27Integral of Discontinuous Functions

Recall that we can’t differentiate a function at the places where it’s discontinuous. However, we can integrate a function even it’s discontinuous. After all, integration of a function y = f(x) from x = a to b means to find the ________________________________________________.

Example 1: Let . Evaluate .

Example 2: Evaluate .

24 , 0( )

2, 0

x xf x

x x

2

2( )f x dx

3.5

0[ ]x dx

Note:Above, we only integrate functions with gaps, not with vertical asymptotes. We can evaluate integrals where the function has vertical asymptotes too. For example,

Page 28Integral of Functions with Symmetries

Recall that a function of y = f(x) may have symmetry with respect to (w.r.t.) the y-axis or the origin.

If a function has symmetry w.r.t. to the y-axis, it’s called an ____ function.

If a function has symmetry w.r.t. to the origin, it’s called an ____ function.

So what?

Knowing the function has one of the two above symmetries can reduce your work on the integral of the type where a > 0.

How so?

Note:

( )a

af x dx