CALCULUS CHAPTER SEVEN NOTES

SECTION 7 – 1 (Day 1) Linear Motion/Displacement/Distance

Linear Motion –

- When a particle is stopped…

- When a particle travels left… - When a particle travels right…

Displacement – called net

- Can be: - 0 –

- Positive –

- Negative –

𝑫𝒊𝒔𝒑𝒍𝒂𝒄𝒆𝒎𝒆𝒏𝒕 = ∫ 𝒅𝒕 𝒃

𝒂

Distance - called total

- Always –

𝑫𝒊𝒔𝒕𝒂𝒏𝒄𝒆 = ∫𝒃

𝒂

𝒅𝒕

Example: A particle moves horizontally along a line with a velocity described by: 𝒗(𝒕) =

𝟔 𝒔𝒊𝒏 𝟑𝒕 𝒇𝒐𝒓 𝟎 ≤ t ≤ 𝝅

𝟐.

6

3

-3

-6

1. Determine when the particle is:

a. Stopped

b. Moving left

c. Moving right

2. What is the particle’s displacement?

3. What is the total distance traveled?

ASSIGNMENT: Page 371 #1, 3, 4, 12 – 16

CALCULUS CHAPTER SEVEN NOTES

SECTION 7 – 1 (Day 2) Linear Motion/Displacement/Distance

This lesson requires the use of graphic calculators:

Example: A particle moves horizontally along the x-axis. Its velocity is described by:

𝒗(𝒕) = 𝒆𝒔𝒊𝒏 𝒕 𝒄𝒐𝒔 𝒕 𝒅𝒕 𝒇𝒐𝒓 𝟎 ≤ 𝒕 ≤ 𝟐𝝅

1

-1

A what values of t is the particle:

a. Stopped

b. Moving left

c. Moving right

What is the displacement?

What is the distance?

Example: The Earth’s acceleration due to gravity is 32 ft/sec2. From ground level, a projectile is fired straight upward with a velocity of 90 ft/sec (initial velocity).

a.) What is the projectile’s velocity after 3 seconds?

b.) When does the projectile hit the ground?

c.) What is the net distance (displacement) when it hits the ground?

d.) What is the total distance travelled?

ASSIGNMENT: Page 371 #5, 8, 9, 10

CALCULUS CHAPTER SEVEN NOTES

SECTION 7 –2 (Day 1) Area Between Curves

Area between Curves: 𝑰𝒇 𝒇(𝒙) ≥ 𝒈(𝒙),𝒕𝒉𝒆𝒏 𝒕𝒉𝒆 𝒂𝒓𝒆𝒂 𝒃𝒆𝒕𝒘𝒆𝒆𝒏 𝒇 𝒂𝒏𝒅 𝒈 𝒃𝒆𝒕𝒘𝒆𝒆𝒏 𝒕𝒉𝒆 𝒗𝒂𝒍𝒖𝒆𝒔 𝒐𝒇 𝒂 𝒂𝒏𝒅 𝒃 𝒊𝒔:

∫ [ − ] 𝒅𝒙𝒃

𝒂

Example: Find the area enclosed by the parabola: y = 2 – x2 and the line: y = -x.

Points of intersection:

Find the Area between the Curves:

Finding area between curves using sub-regions:

Example: Find the area between the curve: y = 4 – x2 and the line y = -x + 2 from x = -2 to x = 3.

ASSIGNMENT: Page 380 -381 #1, #9 (just set up, do not integrate), #13, #24

CALCULUS CHAPTER SEVEN NOTES

SECTION 7 –2 (Day 2) Area Between Curves w/r to y-axis

When finding the area between two curves w/r to the y-axis:

∫[ − ] 𝒅𝒚

Example: Find the area between the curve: x = y2 and the line: x = y + 2.

Points of intersection:

Find the area between the curves:

Other ways of finding areas between curves:

Looking at this area w/r to x:

2 𝒚 = √𝒙

1 𝒚 = 𝒙 − 𝟐

1 2 3 4

ASSIGNMENT: Page 380 – 381 #3, 4, 7, 18, 34a

CALCULUS CHAPTER SEVEN NOTES

SECTION 7 –3 (Day 1) Volume by Cross-sections

The Volume of a Solid – if A(x) is the area of a cross-section from a to b, then:

𝑽 = ∫ 𝒃

𝒂

How to find volume by slicing method:

a. Find whether the cross-section is circular, triangular or a square.

b. Write down the formula for the area of the cross-section

c. Find the limits of integration (a and b)

d. Integrate the area to find the volume.

Example: The base of a solid is a circle of radius of 2. Find the volume of the solid if all cross-sections to the base are squares.

x2 + y2 = 4 cross-section

Solving for y: 𝒚 = ± √𝟒 − 𝒙𝟐 Length of a side:

𝑽 = ∫ (𝒔𝒊𝒅𝒆)𝟐𝟐

−𝟐

𝒅𝒙 =

Example: The solid lies between planes perpendicular to the x-axis at x = -1 and x = 1. The cross sections perpendicular to the x-axis are circular disks whose diameter runs from the parabola y = x2 and the parabola y = 2 – x2. Find the volume.

y

Area of a Circle: A = y = 2 – x2

y = x2

Diameter:

X Radius:

𝑽 = ∫

ASSIGNMENT: Page 390 -391 #1a, 1b, 3, 5, 9

CALCULUS CHAPTER SEVEN NOTES

SECTION 7 –3 (Day 2) Volumes by Disks

Most solids generated are accomplished by rotating a region about the x or y axis. The general formula for finding the volume is as follows:

Using the area of a circle as the cross section: A =

The General formula:

𝑽 = ∫ 𝝅 𝑨(𝒙)𝒅𝒙 = ∫ 𝝅 ( )𝟐𝒃

𝒂

𝒅𝒙 = ∫ 𝝅 ( )𝟐𝒃

𝒂

𝒅𝒙 𝒃

𝒂

Example: Find the volume generated by revolving the region bounded by 𝒇(𝒙) = √𝒙 and the x-axis, for 𝟎 ≤ 𝒙 ≤ 𝟒.

Example: Find the volume generated by revolving the region bounded by 𝒇(𝒙) = 𝒙𝟐 and the x-axis, for 𝟏 ≤ 𝒙 ≤ 𝟑.

ASSIGNMENT: Page 392 #13, 14, 17 – 20

CALCULUS CHAPTER SEVEN NOTES

SECTION 7 –3 (Day 3) Washer Method

If a region revolved to generate a solid does not border or cross the axis of rotation, the solid has a “hole” in it. These cross sections, when sliced look like a “washer”.

Area of Outside Washer:

Area of Inside Washer:

Area of Washer:

General Formula:

𝑽 = ∫ 𝑨𝒓𝒆𝒂 𝒐𝒇 𝑾𝒂𝒔𝒉𝒆𝒓 = 𝒃

𝒂

∫ ( )𝟐𝒃

𝒂

− ( )𝟐 𝒅𝒙 = 𝝅 ∫ ( 𝟐 − 𝟐𝒃

𝒂

)𝒅𝒙

Example: Find the volume of the solid generated by revolving the region bounded by: y = x2 and y = x3 about the x-axis.

1

𝑽 = ∫

1

Example: Find the volume of the solid generated by revolving the region bounded by:

𝒙 = √𝒚 and 𝒙 = 𝒚

𝟐 about the y-axis.

4

3

𝑽 = ∫

2

1

-1 0 1 2

ASSIGNMENT: Page 392 #22 - 26

CALCULUS CHAPTER SEVEN NOTES

SECTION 7 –3 (Day 4) Shell Method

So far we have been rotating equations expressed in a variable rotated about that variable’s axis. Yet, there is another way to calculate volumes when equations are expressed in one variable and are being rotated about the other variable’s axis.

When this is done, a _______________________________ ____________ is formed.

Formula for Shell Method:

𝑽 = ∫ 𝒅𝒙 = ∫ 𝒅𝒙

Example: Find the volume of the solid generated by revolving the region bounded by: y = 2x – x2 and y = x about the y-axis.

1

0 1

𝑽𝒐𝒍𝒖𝒎𝒆 = ∫

ASSIGNMENT: Page 392-393 #39-42, 43a, 44a

CALCULUS CHAPTER SEVEN NOTES

SECTION 7 –3 (Day 4) Moving the Axis of Rotation

Example: Find the volume generated by revolving the region in the first quadrant bounded by y = x3 and y = 4x about the line y = 8.

Adjustments:

Method:

8

6

4

2

0 1 2

ASSIGNMENT: Page 392 #35(a-d), 37c

CALCULUS CHAPTER SEVEN ASSIGNMENTS

SECTION 7 – 1 (Day 1) Linear Motion/Displacement/Distance

ASSIGNMENT: Page 371 #1, 3, 4, 12 – 16

SECTION 7 – 1 (Day 2) Linear Motion/Displacement/Distance

ASSIGNMENT: Page 371 #5, 8, 9, 10

SECTION 7 –2 (Day 1) Area Between Curves

ASSIGNMENT: Page 380 -381 #1, #9 (just set up, do not integrate), #13, #24

SECTION 7 –2 (Day 2) Area Between Curves w/r to y-axis

ASSIGNMENT: Page 380 – 381 #3, 4, 7, 18, 34a

ESSAY AND QUIZ OVER SECTION 7-1 AND 7-2

SECTION 7 –3 (Day 1) Volume by Cross-sections

ASSIGNMENT: Page 390 -391 #1a, 1b, 3, 5, 9

SECTION 7 –3 (Day 2) Volumes by Disks

ASSIGNMENT: Page 392 #13, 14, 17 – 20

SECTION 7 –3 (Day 3) Washer Method

ASSIGNMENT: Page 392 #22 - 26

SECTION 7 –3 (Day 4) Shell Method

ASSIGNMENT: Page 392-393 #39-42, 43a, 44a

SECTION 7 –3 (Day 4) Moving the Axis of Rotation

ASSIGNMENT: Page 392 #35(a-d), 37c

QUIZ OVER SECTION 7-3