chapter 4 · consider a partition 𝑃of the interval [ , ]into 𝑛subintervals by means of points...

CHAPTER 4THE DEFINITE INTEGRAL

4.1 Introduction to Area

Area of Polygons

Area of a Region with a Curved Boundary

What is the area of a circle of radius 1? (Archimedes 287 BC)

Consider the regular inscribed polygons.

And the regular circumscribed polygons.

Sigma Notation

𝑖=1

𝑛

𝑎𝑖 = 𝑎1 + 𝑎2 +⋯+ 𝑎𝑛

Some Examples

1. If σ𝑖=110 𝑎𝑖 = 9 and σ𝑖=1

10 𝑏𝑖 = 7, what are the values of σ𝑖=110 (3𝑎𝑖 − 2𝑏𝑖) and σ𝑖=1

10 (𝑎𝑖 + 4) ?

2. Determine σ𝑖=1𝑛 (𝑎𝑖+1 − 𝑎𝑖).

3. Determine the geometric sum σ𝑘=0𝑛 𝑎𝑟𝑘.

Special Sum Formulas

Area of a Region with a Curved Boundary (revisited)What is the area of a circle of radius 1? (Archimedes 287 BC)

Consider the regular inscribed polygons.

And the regular circumscribed polygons.

Area of a Region Bounded by a Curve of a FunctionCalculate the area of the region under the curve 𝑦 = 𝑥 between 0 and 4.

Consider the region 𝑅 bounded by the parabola 𝑦 = 𝑓 𝑥 = 𝑥2, the 𝑥-axis, and the vertical line 𝑥 = 2.

Calculate its area 𝐴 𝑅 .

4.2 The Definite Integral

Riemann Sums

Consider a function 𝑓 defined on a closed interval [𝑎, 𝑏].

Consider a partition 𝑃 of the interval [𝑎, 𝑏] into 𝑛 subintervals by means of points 𝑎 = 𝑥0 < 𝑥1 < 𝑥2 < ⋯ < 𝑥𝑛−1 < 𝑥𝑛 = 𝑏 and let ∆𝑥𝑖 = 𝑥𝑖 − 𝑥𝑖−1. On each subinterval [𝑥𝑖−1, 𝑥𝑖], pick a sample point ഥ𝑥𝑖 .

The sum 𝑅𝑃 = σ𝑖=1𝑛 𝑓 ഥ𝑥𝑖 ∆𝑥𝑖 is called a Riemann sum for 𝑓

corresponding to the partition 𝑃.

Geometric Interpretation of Riemann Sum

The Definite Integral

Geometric Meaning of Definite Integral

What Functions are Integrable?

Calculating Definite Integral

Evaluate the given definite integrals using definition.

1. 02𝑥 + 1 𝑑𝑥 .

2. 2−1(2x + π)𝑑𝑥 .

Additive Property

Comparison Property

Boundedness Property

Linear Property

4.3 The First Fundamental Theorem of Calculus

Newton, Leibniz, and Calculus

www.calculusbook.net

Two Important Limits

Are these two limits related?

Distance and Velocity

Suppose that an object is traveling along the 𝑥-axis in such away that its velocity at time 𝑡 is 𝑣 = 𝑓 𝑡 feet per second. How far did it travel between 𝑡 = 0 and 𝑡 = 3?

The distance traveled is

lim𝑛→∞

σ𝑖=1𝑛 𝑓 𝑡𝑖 ∆𝑡 0=

3𝑓 𝑡 𝑑𝑡.

What about the distance 𝑠 traveled between 𝑡 = 0 and 𝑡 = 𝑥?

𝑠 𝑥 = න0

𝑥

𝑓 𝑡 𝑑𝑡

What is the derivative of 𝑠?𝑠′(𝑥) = 𝑣 = 𝑓(𝑥)

.

.

First Fundamental Theorem of Calculus

Examples.Find 𝐺′ 𝑥 .

(a) 𝐺 𝑥 = 1𝑥sin 𝑡 𝑑𝑡 (b) 𝐺 𝑥 = 1

𝑥2sin 𝑡 𝑑𝑡

(c) 𝐺 𝑥 = sin 𝑥

co𝑠 𝑥sin 𝑡 𝑑𝑡 (d) 𝐺 𝑥 = 1

𝑥𝑥 sin 𝑡 𝑑𝑡

Evaluating Definite Integral

Let 𝐺 𝑥 = 0𝑥sin 𝑡 𝑑𝑡.

1. Find 𝐺 0 .

2. Let 𝑦 = 𝐺 𝑥 , find 𝑑𝑦

𝑑𝑥.

3. Find the particular solution of the differential equation 𝑑𝑦

𝑑𝑥= sin 𝑥.

4. Use the result in 3. to find 0𝜋sin 𝑡 𝑑𝑡.

4.4 The Second Fundamental Theorem of Calculus

Second Fundamental Theorem of Calculus

Examples.

1. 4−−2

𝑦2 +1

𝑦3𝑑𝑦 .

2. 𝜋/6𝜋/2

2 sin 𝑡 𝑑𝑡 .

Substitution Method

1. 𝑥 𝑥2 + 3−12/7

𝑑𝑥 .

2. 𝑥2 cos 𝑥3 + 5 𝑑𝑥.

3. 14 𝑥−1

3

𝑥𝑑𝑥 .

4. 0𝜋/6

(sin 𝜃)3 cos 𝜃 𝑑𝜃.

5. 01𝑥2 sin 𝑥3

2cos 𝑥3 𝑑𝑥 .

More Examples

𝑓 is a function that has a continuous third derivate. The dashed lines are tangent to the graph 𝑦 = 𝑓(𝑥) at (1,1) and (5,1).

Tell whether the following integrals are positive, negative, or zero.

1. 15𝑓 𝑥 𝑑𝑥.

2. 15𝑓′ 𝑥 𝑑𝑥.

3. 15𝑓′′ 𝑥 𝑑𝑥.

4. 15𝑓′′′ 𝑥 𝑑𝑥.

Water leaks out of a 55-gallon tank at the rate 𝑉′ 𝑡 = 11 − 1.1𝑡 where 𝑡 is measured in hours and 𝑉 in gallons. Initially the tank is full.1. How much water leaks out of the tank between 𝑡 = 3 and 𝑡 = 5 hours?2. How long does it take until there are just 5 gallons remaining in the tank?

4.5 The Mean Value Theorem for Integrals and the Use of Symmetry

Average Value of a Function

Do you still remember the Mean Value Theorem for Derivative?

If 𝑓 is integrable on the interval [𝑎, 𝑏], then the average value of 𝑓 on [𝑎, 𝑏] is:

If you consider the definite integral from over [𝑎, 𝑏] to be the area between the curve 𝑓(𝑥) and the 𝑥-axis, 𝑓𝑎𝑣𝑒 is the height of the rectangle that would be formed over that same interval containing precisely the same area.

b

a

ave dxxfab

f )(1

The Mean Value Theorem for Integrals

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

Example.

1. Suppose the temperature in degrees Celsius of a metal bar of length 2meters depends on the position 𝑥 according to the function 𝑇(𝑥) =40 + 20𝑥(2 − 𝑥). Find the average temperature in the bar. Is there a point where the actual temperature equals the average temperature.

2. Find all values of c that satisfy the Mean Value Theorem for 𝑓(𝑥) =|𝑥| on [−2,2].

b

a

dttfab

cf )(1

)(

Symmetry Theorem

If 𝑓 is an even function then

If 𝑓 is an odd function, then

Periodicity

If 𝑓 is periodic with period 𝑝, then

Examples. Evaluate

1. .

2. .

3. .

4.6 Numerical Integration

Approximation of Definite Integral

If 𝑓 is continuous on a closed interval [𝑎, 𝑏], then the definite integral must exist. However, it is not always easy or possible to find the definite integral.

Examples.

නsin 𝑥2 𝑑𝑥

නsin 𝑥

𝑥𝑑𝑥

In these cases, we use other methods to closely approximate the definite integral.

Methods

1. Left (or right or midpoint) Riemann sums

Estimate the area with rectangles

2. Trapezoidal Rule

Estimate the area with several trapezoids

3. Simpson’s Rule

Estimate the area with the region contained under several parabolas

Left Riemann Sum

න𝑎

𝑏

𝑓 𝑥 𝑑𝑥 ≈ 𝑓 𝑥0 + 𝑓 𝑥1 +⋯+ 𝑓(𝑥𝑛−1) ∆𝑥, ∆𝑥 =𝑏 − 𝑎

𝑛

𝐸𝑛 =𝑏 − 𝑎 2

2𝑛𝑓′ 𝑐 , for 𝑎 ≤ 𝑐 ≤ 𝑏

Right Riemann Sum

න𝑎

𝑏

𝑓 𝑥 𝑑𝑥 ≈ 𝑓 𝑥1 + 𝑓 𝑥2 +⋯+ 𝑓(𝑥𝑛) ∆𝑥, ∆𝑥 =𝑏 − 𝑎

𝑛

𝐸𝑛 = −𝑏 − 𝑎 2

2𝑛𝑓′ 𝑐 , for 𝑎 ≤ 𝑐 ≤ 𝑏

Midpoint Riemann Sum

න𝑎

𝑏

𝑓 𝑥 𝑑𝑥 ≈ 𝑓𝑥0 + 𝑥1

2+ 𝑓

𝑥1 + 𝑥22

+⋯+ 𝑓𝑥𝑛−1 + 𝑥𝑛

2∆𝑥, ∆𝑥 =

𝑏 − 𝑎

𝑛

𝐸𝑛 =𝑏 − 𝑎 3

24𝑛2𝑓" 𝑐 , for 𝑎 ≤ 𝑐 ≤ 𝑏

Trapezoidal Rule

න𝑎

𝑏

𝑓 𝑥 𝑑𝑥 ≈∆𝑥

2𝑓 𝑥0 + 2𝑓 𝑥1 + 2𝑓 𝑥2 +⋯+ 2𝑓(𝑥𝑛−1) + 𝑓(𝑥𝑛) , ∆𝑥 =

𝑏 − 𝑎

𝑛

𝐸𝑛 = −𝑏 − 𝑎 3

12𝑛2𝑓" 𝑐 , for 𝑎 ≤ 𝑐 ≤ 𝑏

Simpson Rule (for even 𝑛)

න𝑎

𝑏

𝑓 𝑥 𝑑𝑥 ≈∆𝑥

3𝑓 𝑥0 + 4𝑓 𝑥1 + 2𝑓 𝑥2 + 4𝑓 𝑥3 +⋯+ 2𝑓(𝑥𝑛−2) + 4𝑓(𝑥𝑛−1) + 𝑓(𝑥𝑛) , ∆𝑥 =

𝑏 − 𝑎

𝑛

𝐸𝑛 = −𝑏 − 𝑎 5

180𝑛4𝑓(4) 𝑐 , for 𝑎 ≤ 𝑐 ≤ 𝑏

Examples

1. Approximate 13 1

1+𝑥2𝑑𝑥 by using left Riemann sum, trapezoidal rule, and

Simpson rule with 𝑛 = 4. Then determine a maximum of the absolute error.

2. Determine 𝑛 so that the trapezoidal rule will approximate 13 1

𝑥𝑑𝑥 with an error

𝐸𝑛 satisfying |𝐸𝑛| ≤ 0.01.

3. Determine 𝑛 so that the Simpson rule will approximate 13 1

𝑥𝑑𝑥 with an error

𝐸𝑛 satisfying |𝐸𝑛| ≤ 0.01.

4. On her way to work, Ani noted her speed every 3 minutes. The results are shown in the table below. How far did she drive?