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Lecture Comments (3)

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Post by Steve Denton on October 22, 2012

At 21:30 in example 4, how exactly did the 1/2 cancel with the division by 2 in the next term?

1 answer

Last reply by: Jingwei Xie
Mon Apr 21, 2014 10:06 PM

Post by Steve Denton on October 22, 2012

At 14:40 or so, isn't the height of an isosceles triangle s(root3)/2 and the area then, not s^2, but s^2(root3)/4?

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Revolving Solids Known Cross Sections

Finding volume of solid with known cross sections and bounds

Find area of known cross-section
Substitute bounding functions to variables in area equation appropriately
Evaluate integral of area of known cross-section

Knowing derivations of disk, washer, and cylindrical formulas dramatically improves understanding of known cross-sections volumes

Revolving Solids Known Cross Sections

Find the volume of a solid whose base is defined by the region bounded by y = x, y = 0, and x = 1 with the diameters of semi-circle cross-sections along the y-axis.

Area of a semi-circle = [1/2] πr²
V = [1/2] π∫₀¹ ([x/2])² dx
V = [(π)/8] [(x³)/3] |₀¹

V = [(π)/24]

Find the volume of a solid whose base is defined by the region bounded by y = sinx and y = 0 from x = 0 to x = π with the square cross-sections.

A = l²
V = ∫₀^π y² dx
V = ∫₀^π sin² x dx
V = ∫₀^π ([(1 − cos2x)/2]) dx
V = [1/2] ∫₀^π 1 − cos2x dx
V = [1/2] (∫₀^π dx − ∫₀^π cos2x dx)
u = 2x
du = 2 dx
V = [1/2] (x |₀^π − [1/2] sin2x |₀^π)

V = [(π)/2]

Find the volume of a solid whose base is defined by the region bounded by y = x² and y = x with equilateral triangle cross-sections along the y-axis.

Area of equilateral triangle = [(√3)/4] l²
Intersection points
x² = x
x = 0, 1
In between 0 and 1, x ≥ x²
l = x − x²
V = [(√3)/4] ∫₀¹ (x − x²)² dx
V = [(√3)/4] ∫₀¹ x⁴ − 2x³ + x² dx
V = [(√3)/4] ([(x⁵)/5] − [(x⁴)/2] + [(x³)/3]) |₀¹
V = [(√3)/4] ([1/5] − [1/2] + [1/3] − 0)

V = [(√3)/120]

Find the volume of a solid whose base is defined by the region bounded by y = x² and y = x with square cross-sections along the x-axis.

This time, we're taking triangles parallel to the x-axis
We need to rewrite both of the equations in terms of x
x² = y
x = ±√y
The intersection is in the first quadrant, so we only care about the positive value
x = √y
Intersects at y = 0, 1
A = l²
l = √y − y
V = ∫₀¹ (√y − y)² dy
V = ∫₀¹ (y − 2y√y + y²) dy
V = ∫₀¹ (y − 2 y^[3/2] + y²) dy
V = [(y²)/2] − [(4 y^[5/2])/5] + [(y³)/3] |₀¹
V = [1/2] − [4/5] + [1/3]

V = [1/30]

Find the volume of a solid whose base is defined by the region bounded by y = 0, y = 2, x = 0, and x = 6 with semi-circle cross-sections along the x-axis.

The area bound by those lines is simply a square, with lines parallel to the axes.
A = [1/2] πr²
r = [6/2]
V = [(π)/2] ∫₀² 3² dy
V = [(π)/2] 9x |₀²

V = 9π

Find the volume of a solid whose base is defined by the region bounded by y = x² + 1 and y = −x² + 3 with semi-circle cross-sections along the y-axis.

Find their intersection points
x² + 1 = −x² + 3
2x² = 2
x = −1, 1
A = [1/2] πr²
r = [((−x² + 3) − (x² + 1))/2]
V = [(π)/2] ∫₋₁¹ (−x² + 1)² dx
V = [(π)/2] ∫₋₁¹ x⁴ − 2x² + 1 dx
V = [(π)/2] ([(x⁵)/5] − [(2x³)/3] + x) |₋₁¹
V = [(π)/2] ([1/5] − [2/3] + 1 − ([(−1)/5] − [(−2)/3] − 1))
V = [(π)/2] ([1/5] − [2/3] + 1 + [1/5] − [2/3] + 1)

V = [(8π)/15]

Find the volume of a solid whose base is defined by the region bounded by y = cosx and y = 0 from x = 0 to x = [(π)/2] with square cross-sections along the y-axis.

V = ∫₀^[(π)/2] cos² x dx
V = ∫₀^[(π)/2] [(1 + cos2x)/2] dx
V = [1/2] (x + [1/2] sin2x) |₀^[(π)/2]
V = [1/2] ([(π)/2] + 0)

V = [(π)/4]

Find the volume of a solid whose base is defined by the region bounded by y = √{1 − x²} and y = 0 with semi-circle cross-sections along the y-axis.

V = [(π)/2] ∫₋₁¹ ([(√{1 − x²})/2])² dx
V = [(π)/8] ∫₋₁¹ 1 − x² dx
We can use symmetry
V = [(π)/4] ∫₀¹ 1 − x² dx
V = [(π)/4] (x − [(x³)/3]) |₀¹
V = [(π)/4] (1 − [1/3] − 0)

V = [(π)/6]

Find the volume of a solid whose base is defined by the region bounded by x = y³ and x = y² with square cross-sections along the x-axis.

V = ∫₀¹ (y² − y³)² dy
V = ∫₀¹ y⁴ − 2 y⁵ + y⁶ dy
V = ([(y⁵)/5] − [(y⁶)/3] + [(y⁷)/7]) |₀¹
V = [1/5] − [1/3] + [1/7]

V = [1/105]

Find the volume of a solid whose base is defined by the region bounded by x = y³ and x = y² with square cross-sections along the y-axis.

Rewrite in terms of y
y = ³√{x} and y = ±√x
First quadrant again, so we're only interested in the positive
V = ∫₀¹ (√x − ³√{x})² dx
V = ∫₀¹ x − 2 x^[5/6] + x^[2/3] dx
V = ([(x²)/2] − [(12 x^[11/6])/11] + [(3 x^[5/3])/5]) |₀¹
V = [1/2] − [12/11] + [3/5]

V = [1/110]

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Answer

Revolving Solids Known Cross Sections

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

Intro

Revolving Solids Known Cross Sections

Example 1

Example 2

Example 3

Example 4

Example 5

Intro 0:00
Revolving Solids Known Cross Sections 0:08

Example 1

Example 2 6:01
Example 3 11:03
Example 4 17:29
Example 5 22:19

Mathematics: AP Calculus AB

Section 1: Functions
	Definitions & Properties of Functions	11:26
	Graphing	13:58
	Inverse Functions	6:47
	Polynomial Functions	5:04
	Trigonometric Functions	6:45
	Inverse Trigonometric Functions	5:58
	Trigonometric Identities	17:42
	Exponential Functions	5:53
	Logarithmic Functions	7:08
	Rational Functions	15:36
	Conic Sections	14:58
Section 2: Limits and Continuity
	Limit Definition & Properties	7:15
	Solving Limits with Algebra	8:01
	Rational Limit Rules	3:16
	One Sided Limits	9:57
	Special Trigonometric Limits	8:28
	Limits & Continuity	10:14
	Limits: Multiple Choice Practice	6:16
Section 3: Derivatives
	Derivative Definition & Properties	4:11
	Basic Rules of Differentiation	7:07
	Power Rule	7:14
	Trigonometric Rules	7:53
	Product Rule	11:11
	Quotient Rule	16:50
	Chain Rule	19:48
	Higher Order Derivatives	15:00
	Derivatives of Exponential Functions	13:14
	Derivatives of Logarithmic Functions	11:30
	Derivatives of Inverse Trigonometric Functions	16:54
	Implicit Differentiation	16:53
	Multiple Choice Practice: Derivatives	11:07
Section 4: Applications of Derivatives
	Tangent & Normal Lines	22:36
	Position Velocity & Acceleration	18:42
	Related Rates	26:22
	Minimum & Maximum	12:22
	Concavity	11:43
	Rolles Theorem	8:28
	Mean Value Theorem	9:39
	Differentials	12:25
	Applications of Derivatives: Multiple Choice Practice	13:21
	Applications of Derivatives: Free Response Practice	10:22
Section 5: Integrals
	Definition of Integrals	1:08
	Integrals of Power Rule	8:50
	Integrals Basic Rules of Integration	9:43
	Trigonometric Rules of Integrals	8:58
	Chain Rule	13:59
	Integrals of Exponential Functions	12:52
	Integrals of Natural Logarithmic Functions	13:00
	Integrals of Inverse Trigonometric Functions	8:29
	Integrals: Multiple Choice Practice	15:37
Section 6: Applications of Integrals
	Fundamental Theorem of Calculus	15:55
	Area Under A Curve	18:34
	Reimann Sums	10:35
	Trapezoid Rule	12:46
	Mean Value Theorem	11:22
	Second Fundamental Theorem of Calculus	4:44
	Area Between Curves	16:39
	Revolving Solids Washer Disk Methods	21:09
	Revolving Solids Cylindrical Shells Method	26:46
	Revolving Solids Known Cross Sections	27:41
	Differential Equations Eulers Method	17:54
	Differential Equations Slope Fields	16:30
	Application of Integrals: Multiple Choice Practice	14:19
	Application of Integrals: Free Response Practice	9:14
Section 7: Sample AP Test
	AP Calculus AB Practice test: Section 1: Multiple Choice Part 1	17:50
	AP Calculus AB Practice test: Section 1: Multiple Choice Part 2	17:32
	AP Calculus AB Practice test: Section 1: Multiple Choice Part 3	22:14
	AP Calculus AB Practice test: Section 1: Multiple Choice Part 4	19:35
	AP Calculus AB Practice test: Section 1: Multiple Choice Part 5	25:43
	AP Calculus AB Practice Test: Section 2: Free Response Part 1	16:50
	AP Calculus AB Practice Test: Section 2: Free Response Part 2	21:36
	AP Calculus AB Practice Test: Section 2: Free Response Part 3	13:39

Related Books

	Cracking the AP Calculus AB Exam, 2015 Edition Author: Princeton Review ISBN: 804124809 Publisher: Princeton Review Year: 2014 The book includes 3 full length practice tests with detailed explanations, a review of all the key concepts, and targeted strateeies to ace th exam for your highest score.
	Cliffs AP: Calculus AB & BC, 3rd Edition Authors: Dale W Johnson, Kerry J King ISBN: 764586831 Publisher: Cliffs Notes Year: 2001 The book has a topic-by topic breakdown and lots of problem approach suggestions for both free response and multiple choice calculus questions. The book also includes two full length tests.

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Revolving Solids Known Cross Sections

Slide Duration:

Table of Contents

Section 1: Functions

Definitions & Properties of Functions

11m 26s

Intro