Appropriate Integrals


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Lecture Notes

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Section 2, Page 1 to page 2

Method for finding the area between two curves. Includes worked example of finding the area bounded by the curve y=x(x2-3) and a horizontal tangent line.

Instructor: Prof. Jason Starr
Prior Knowledge: Differentials (section 1 of lecture 13) and Fundamental Theorem of Calculus (section 3 of lecture 15)
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Section 3, Page 3 to page 4

Introduces the disk method with worked examples of finding the volume of a right circular cone and a sphere.

Instructor: Prof. Jason Starr
Prior Knowledge: Differentials (section 1 of lecture 13) and Fundamental Theorem of Calculus (section 3 of lecture 15)
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Section 4, Page 4

Generalization of the disk method when cross-sectional areas are known. Includes worked example.

Instructor: Prof. Jason Starr
Prior Knowledge: The Disk Method (section 3 of this lecture)
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Section 5, Page 5 to page 6

Variation of disk method using the difference of two disks to create washers. Includes worked example of finding the volume of material of a dog dish.

Instructor: Prof. Jason Starr
Prior Knowledge: The Disk Method (section 3 of this lecture)
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Section 1, Page 1 to page 2

Derivation of average value formula using Reimann sums. Example of finding the average radius r(x) = r0 + Acos(wx) of a wire made by a vibrating machine.

Instructor: Prof. Jason Starr
Prior Knowledge: Riemann Sum (section 3 of lecture 14) and Fundamental Theorem of Calculus (section 3 of lecture 15)
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Section 3, Page 5 to page 7

Explanation of the shell method as an alternative to the disk and washer methods. Revisits worked example of finding the volume of material of a dog dish (previously solved using the washer method in section 5 of lecture 19).

Instructor: Prof. Jason Starr
Prior Knowledge: Volumes of Solids of Revolution (sections 3 to 5 of lecture 19)
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Section 3, Page 3 to page 6

Derivation of formula for finding the length of a curve. Includes worked examples. Example 2 demonstrates finding the length of a curve with equation y = f(x) by changing to parametric equations.

Instructor: Prof. Jason Starr
Prior Knowledge: Parametric Equations (section 1 of this lecture)
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Section 1, Page 1 to page 2

Introduces surface area of a surface of revolution using the case of a right circular cone.

Instructor: Prof. Jason Starr
Prior Knowledge: None
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Section 2, Page 2 to page 5

Formula for the surface area of a surface of revolution. Includes examples of a line segment, a semicircle, and an astroid.

Instructor: Prof. Jason Starr
Prior Knowledge: Parametric Equations and Arc Length (sections 1 and 3 of lecture 21)
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Section 3, Page 4 to page 6

Method for finding arc length of a polar curve, including example of a cardioid. Method for finding surface area of a surface of revolution, with example of a cardioid used to approximate the surface area of an apple.

Instructor: Prof. Jason Starr
Prior Knowledge: Arc Length (section 3 of lecture 21), Surface Area of Surface of Revolution (section 2 of lecture 22), and Polar Coordinate Curves (section 3 of lecture 22)
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Section 5, Page 6 to page 7

Method for finding the area of a region bounded by a polar curve, using the example of a cardioid.

Instructor: Prof. Jason Starr
Prior Knowledge: Arc Length in Polar Coordinates (section 3 of this lecture)
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Section 3, Page 3 to page 4

Problems and answers without full explanation. Finding tangent lines to an ellipse, minimizing surface area of a grain silo, finding the volume of a solid of revolution, computing an antiderivative using trig substitution, and computing an antiderivative using integration by parts.

Instructor: Prof. Jason Starr
Prior Knowledge: Tangent Lines (section 1 of lecture 2), Max/Min Problems (section 2 of lecture 10), Volume of Solids of Revolution (section 3 of lecture 19), Inverse Substitution (section 3 of lecture 25), Integration by Parts (section 1 of lecture 27)
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Section 1, Page 1 to page 2

Finding the average value of a function over an interval, with diagrams and examples relating to temperature, alternating current, and chords in a unit circle.

Instructor: Prof. David Jerison
Prior Knowledge: None
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Online Textbook Chapter

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Definition as a method for finding the area of a volume under a surface defined by a function of x and y.

Instructor: Prof. Daniel J. Kleitman
Prior Knowledge: Riemann Sums (OT20.2)
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Practice Problems

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Problem 3 (page 2)

Proving the volume of a cone using the washer method for finding volumes of solids of revolution.

Instructor: Prof. Jason Starr
Prior Knowledge: None
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Problem 2 (page 2)

Sketching and computing the area of the polar curve r = cos(3*θ).

Instructor: Prof. Jason Starr
Prior Knowledge: None
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Exam Questions

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Problem 1 (page 2)

Finding the volume of a solid of revolution about the x-axis.

Instructor: Prof. Jason Starr
Prior Knowledge: None
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Problem 2 (page 3)

Finding the volume of a solid of revolution about the y-axis.

Instructor: Prof. Jason Starr
Prior Knowledge: None
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Problem I.1 (page 1) to problem IV.5 (page 4)

Eighteen problems with answers but not complete solutions on these four topics.

Instructor: Prof. Jason Starr
Prior Knowledge: None
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Problem 8 (page 1)

Finding the volume of a solid of revolution about the x-axis.

Instructor: Prof. Jason Starr
Prior Knowledge: None
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Problem 10 (page 1)

Finding the area of the region in the 1st and 3rd quadrants between two circles defined in polar coordinates.

Instructor: Prof. Jason Starr
Prior Knowledge: None
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Problem 5.1 (page 3) to Problem 5.2 (page 4)

Two problems which involve finding the area between a curve and a tangent line and the area between two parabolas.

Instructor: Prof. Jason Starr
Prior Knowledge: None
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Problem 5.3 (page 4)

Finding the volume of a solid of revolution about the x-axis.

Instructor: Prof. Jason Starr
Prior Knowledge: None
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Problem 5.4 (page 4)

Finding the arc length of a segment of the graph of the natural logarithm.

Instructor: Prof. Jason Starr
Prior Knowledge: None
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Problem 3 (page 1)

Setting up a definite integral for the amount of money in an account at the end of a year.

Instructor: Prof. David Jerison
Prior Knowledge: None
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Problem 6 (page 1)

Finding the volume of a glass vase in the shape of a solid of revolution

Instructor: Prof. David Jerison
Prior Knowledge: None
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Problem 7 (page 1)

Finding the volume of ice cream in an overfilled cone defined by a solid of revolution.

Instructor: Prof. David Jerison
Prior Knowledge: None
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Problem 8 (page 1)

Finding the average value of rectangles inscribed at random in a quarter of a circle.

Instructor: Prof. David Jerison
Prior Knowledge: None
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Problem 2 (page 1)

Finding the volume of candy needed to fill a Great Pumpkin with shape defined in terms of a solid of revolution.

Instructor: Prof. David Jerison
Prior Knowledge: None
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Problem 4 (page 1)

Finding the average area of slices of a SmartHam defined in terms of a solid of revolution.

Prior Knowledge: None
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Problem 3 (page 1)

Finding the volume of a solid of revolution about the y-axis.

Instructor: Prof. David Jerison
Prior Knowledge: None
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Problem 4 (page 1)

Finding the average amount score and average amount of sleep students get the night before a test.

Instructor: Prof. David Jerison
Prior Knowledge: None
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Problem 4 (page 1)

Finding the volume of a solid of revolution about the y-axis.

Instructor: Prof. David Jerison
Prior Knowledge: None
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Problem 5 (page 1)

Sketching a curve defined in polar coordinates and finding the area inside it.

Instructor: Prof. David Jerison
Prior Knowledge: None
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Problem 6 (page 1)

Finding the length of a curve defined parametrically.

Instructor: Prof. David Jerison
Prior Knowledge: None
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Problem 7 (page 1)

Setting up an integral for the length of one arc of the sine curve, and estimating the value of the integral.

Instructor: Prof. David Jerison
Prior Knowledge: None
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Problem 8 (page 1)

Setting up and evaluating an integral for the mass of a disc with variable mass density.

Instructor: Prof. David Jerison
Prior Knowledge: None
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Problem 2 (page 1)

Finding the volume of a solid of revolution about the y-axis.

Instructor: Prof. David Jerison
Prior Knowledge: None
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Problem 4 (page 1)

Setting up and estimating the value of an integral representing the length of a given curve.

Instructor: Prof. David Jerison
Prior Knowledge: None
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Problem 6 (page 1)

Setting up and evaluating an integral to represent the uncovered area of the two moons involved in a lunar eclipse on another planet.

Instructor: Prof. David Jerison
Prior Knowledge: None
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Problem 4 (page 1)

Setting up an integral to find the length of a curve given in rectangular coordinates.

Instructor: Prof. David Jerison
Prior Knowledge: None
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Problem 5 (page 1) to problem (page 1)

Sketching a spiral defined in polar coordinates, counting the times it crosses the x-axis, and finding the area of specific regions of the spiral.

Instructor: Prof. David Jerison
Prior Knowledge: None
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Problem 8 (page 1)

Finding the amount of wine that can be held in a glass defined in terms of a solid of revolution about the y-axis.

Instructor: Prof. David Jerison
Prior Knowledge: None
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Problem 11 (page 2)

Setting up an integral for the length of the main cables in a suspension bridge, and using it to find the average length of the vertical cables connecting to the roadway.

Instructor: Prof. David Jerison
Prior Knowledge: None
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Problem 14 (page 2)

Sketching a curve given in polar coordinates and finding the area swept by a line segment as one of the endpoints moves along this curve.

Instructor: Prof. David Jerison
Prior Knowledge: None
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Analysis of Graphs Limits of Functions Asymptotic & Unbounded Behavior Continuity: Property of Functions Parametric, Polar & Vector Function Concept of the Derivative Derivative at a Point Derivative as a Function Second Derivatives Applications of Derivatives Computation of Derivatives Interpretations & Properties of Definite Integrals Applications of Integrals Fundamental Theorem of Calculus Applications of Antidifferentiation Numerical Approximations to Definite Integ Concept of Series Series of Constants Taylor Series Graphs of Functions Intuitive Understanding: Limiting Process Calculating Limits Using Algebra Asymptotic Behavior: Infinity Continuity Continuity in Terms of Limits Analysis of Planar Curves Derivative Presented Graphically, Numerically & Analytically Derivative Interpreted as Instantaneous Rate of Change Derivative: Limit of the Difference Quotient Differentiability & Continuity Tangent Line - Curve at a Point & Local Linear Approximation Instantaneous Rate of Change Approximate Rate of Change Characteristics of f & f Relationship Between Behavior of f & Sign of f Mean Value Theorem & Geometric Consequences Characteristics: Graphs of f, f & f Relationship Between Concavity of f & Sign of f'' Points of Inflection - Concavity Changes Analysis of Curves Applications of Derivatives Optimization: Absolute & Relative Extrema Modeling Rates of Change Implicit Differentiation & Derivatives of Inverse Functions Derivative as a Rate of Change l'Hôpital's Rule Geometric Interpretation of Differential Equations Derivatives of Basic Functions Rules: Derivative of Sums, Products & Quotients of Functions Chain Rule & Implicit Differentiation Derivatives: Parametric, Polar & Vector Functions Definite Integral - Limit of Riemann Sums Rate of Change of a Quantity/Change of Quantity over Interval Properties of Definite Integrals Appropriate Integrals Evaluate Definite Integrals Representation of Specific Antiderivatives Antiderivatives From Derivatives of Basic Functions Integration by Substitution, Parts & Partial Fractions Improper Integrals Separable Equations & Modeling Riemann & Trapezoidal Sums Series, Convergence, Divergence Motivating Examples Geometric Series with Applications The Harmonic Series Alternating Series - Error Bound Series as Riemann Sums & the Integral Test Ratio Test - Convergence/Divergence Comparison Test Taylor Polynomial Approximation Maclaurin & Taylor Series Maclaurin Series for Basic Functions Manipulating Taylor Series Power Series Defining Functions Radius/Interval of Convergence Lagrange Error Bound