# Appropriate Integrals

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## Lecture Notes

#### Areas Between Curves

PDF
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)

#### Volumes of Solids of Revolution: The Disk Method

PDF
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)

#### The Slice Method

PDF
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)

#### Volumes of Solids of Revolution: The Washer Method

PDF
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)

#### Average Value of a Function

PDF
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)

#### Volumes of Solids of Revolution: The Shell Method

PDF
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)

#### Arc Length

PDF
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)

#### Surface Area of a Right Circular Cone

PDF
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

#### Surface Area of a Surface of Revolution

PDF
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)

#### Arc Length in Polar Coordinates

PDF
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)

#### Area of a Region Enclosed by a Polar Curve

PDF
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)

#### Review Problems

PDF
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)

#### Average Value

PDF
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

## Online Textbook Chapter

#### Area Integrals

Document

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)

## Practice Problems

#### Washer Method

PDF
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

#### Three-Leaved Rose

PDF
Problem 2 (page 2)

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

Instructor: Prof. Jason Starr
Prior Knowledge: None

## Exam Questions

#### Solids of Revolution: X-Axis

PDF
Problem 1 (page 2)

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

Instructor: Prof. Jason Starr
Prior Knowledge: None

#### Solids of Revolution: Y-Axis

PDF
Problem 2 (page 3)

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

Instructor: Prof. Jason Starr
Prior Knowledge: None

#### Miscellaneous Integration Problems

PDF
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

#### Solids of Revolution

PDF
Problem 8 (page 1)

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

Instructor: Prof. Jason Starr
Prior Knowledge: None

#### Area of a Lune

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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

#### Area Between Two Curves

PDF
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

#### Volume of Solid of Revolution

PDF
Problem 5.3 (page 4)

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

Instructor: Prof. Jason Starr
Prior Knowledge: None

#### Arc Length of a Segment of a Curve

PDF
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

#### Continuously Compounded Interest

PDF
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

#### Solids of Revolution: Volume of A Vase

PDF
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

#### Solids of Revolution: Ice Cream Cone

PDF
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

#### Average Value of a Function

PDF
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

#### Solids of Revolution: Great Pumpkin

PDF
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

#### Average Value: SmartHam

PDF
Problem 4 (page 1)

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

Prior Knowledge: None

#### Solid of Revolution

PDF
Problem 3 (page 1)

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

Instructor: Prof. David Jerison
Prior Knowledge: None

#### Average Value: Test Scores

PDF
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

#### Solids of Revolution

PDF
Problem 4 (page 1)

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

Instructor: Prof. David Jerison
Prior Knowledge: None

#### Area Inside a Polar Curve

PDF
Problem 5 (page 1)

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

Instructor: Prof. David Jerison
Prior Knowledge: None

#### Arc Length of a Curve

PDF
Problem 6 (page 1)

Finding the length of a curve defined parametrically.

Instructor: Prof. David Jerison
Prior Knowledge: None

#### Estimating the Length of the Sine Curve

PDF
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

#### Mass of a Metal Disc

PDF
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

#### Solids of Revolution

PDF
Problem 2 (page 1)

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

Instructor: Prof. David Jerison
Prior Knowledge: None

#### Arc Length of a Curve

PDF
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

#### Integrals in Polar Coordinates: Lunar Eclipse

PDF
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

#### Arc Length of a Curve

PDF
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

#### Polar Coordinates: Spiral

PDF
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

#### Solids of Revolution: Wine Glass

PDF
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

#### Average Value: Suspension Bridge

PDF
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