- Areas Between Curves
- Volumes of Solids of Revolution: The Disk Method
- The Slice Method
- Volumes of Solids of Revolution: The Washer Method
- Average Value of a Function
- Volumes of Solids of Revolution: The Shell Method
- Arc Length
- Surface Area of a Right Circular Cone
- Surface Area of a Surface of Revolution
- Arc Length in Polar Coordinates
- Area of a Region Enclosed by a Polar Curve
- Review Problems
- Average Value
- Area Integrals
- Washer Method
- Three-Leaved Rose
- Solids of Revolution: X-Axis
- Solids of Revolution: Y-Axis
- Miscellaneous Integration Problems
- Solids of Revolution

- Area of a Lune
- Area Between Two Curves
- Volume of Solid of Revolution
- Arc Length of a Segment of a Curve
- Continuously Compounded Interest
- Solids of Revolution: Volume of A Vase
- Solids of Revolution: Ice Cream Cone
- Solids of Revolution: Great Pumpkin
- Average Value: SmartHam
- Average Value: Test Scores
- Area Inside a Polar Curve
- Arc Length of a Curve
- Estimating the Length of the Sine Curve
- Mass of a Metal Disc
- Integrals in Polar Coordinates: Lunar Eclipse
- Polar Coordinates: Spiral
- Solids of Revolution: Wine Glass
- Average Value: Suspension Bridge
- Polar Curves

Method for finding the area between two curves. Includes worked example of finding the area bounded by the curve y=x(x^{2}-3) and a horizontal tangent line.

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

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

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

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

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

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.

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

Derivation of average value formula using Reimann sums. Example of finding the average radius r(x) = r_{0} + Acos(wx) of a wire made by a vibrating machine.

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

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

18.01

*Single Variable Calculus*, Fall 2006

Prof. David Jerison

**Course Material Related to This Topic:**

- Complete exam problem 8 on page 1
- Check solution to exam problem 8 on page 1

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).

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

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.

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

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

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

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

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

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.

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

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

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

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.

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

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.

18.01

*Single Variable Calculus*, Fall 2006

Prof. David Jerison

**Course Material Related to This Topic:**

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

18.013A

*Calculus with Applications*, Spring 2005

Prof. Daniel J. Kleitman

**Course Material Related to This Topic:**

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

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

- Complete practice problem 3 on page 2
- Check solution to practice problem 3 on page 5

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

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

- Complete practice problem 2 on page 2
- Check solution to practice problem 2 on pages 5–6

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

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

- Complete exam problem 1 on page 2
- Check solution to exam problem 1 on page 1

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

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

- Complete exam problem 2 on page 3
- Check solution to exam problem 2 on page 2

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

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

- Complete exam problem I.1 on page 1 to problem IV.5 on page 4
- Check solution to exam problems on page 4

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

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

- Complete exam problem 8 on page 1
- Check solution to exam problem 8 on page 3

18.01

*Single Variable Calculus*, Fall 2006

Prof. David Jerison

**Course Material Related to This Topic:**

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

- Complete exam problem 3 on page 1
- Check solution to exam problem 3 on page 1
- Complete exam problem 4 on page 1
- Check solution to exam problem 4 on page 1

- Complete exam problem 2 on page 1
- Check solution to exam problem 2 on page 1

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

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

- Complete exam problem 10 on page 1
- Check solution to exam problem 10 on page 4

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

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

- Complete exam problem 5.1 on page 3 to problem 5.2 on page 4
- Check solution to exam problems on page 4

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

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

- Complete exam problem 5.3 on page 4
- Check solution to exam problem on page 4

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

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

- Complete exam problem 5.4 on page 4
- Check solution to exam problem on page 4

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

18.01

*Single Variable Calculus*, Fall 2006

Prof. David Jerison

**Course Material Related to This Topic:**

- Complete exam problem 3 on page 1
- Check solution to exam problem 3 on page 1

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

18.01

*Single Variable Calculus*, Fall 2006

Prof. David Jerison

**Course Material Related to This Topic:**

- Complete exam problem 6 on page 1
- Check solution to exam problem 6 on page 1

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

18.01

*Single Variable Calculus*, Fall 2006

Prof. David Jerison

**Course Material Related to This Topic:**

- Complete exam problem 7 on page 1
- Check solution to exam problem 7 on page 1

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

18.01

*Single Variable Calculus*, Fall 2006

Prof. David Jerison

**Course Material Related to This Topic:**

- Complete exam problem 2 on page 1
- Check solution to exam problem 2 on page 1

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

18.01

*Single Variable Calculus*, Fall 2006

Prof. David Jerison

**Course Material Related to This Topic:**

- Complete exam problem 4 on page 1
- Check solution to exam problem 4 on page 1

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

18.01

*Single Variable Calculus*, Fall 2006

Prof. David Jerison

**Course Material Related to This Topic:**

- Complete exam problem 4 on page 1
- Check solution to exam problem 4 on page 1

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

18.01

*Single Variable Calculus*, Fall 2006

Prof. David Jerison

**Course Material Related to This Topic:**

- Complete exam problem 5 on page 1
- Check solution to exam problem 5 on page 1

18.01

*Single Variable Calculus*, Fall 2006

Prof. David Jerison

**Course Material Related to This Topic:**

Finding the length of a curve defined parametrically.

- Complete exam problem 6 on page 1
- Check solution to exam problem 6 on page 1

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

- Complete exam problem 4 on page 1
- Check solution to exam problem 4 on page 1

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

- Complete exam problem 4 on page 1
- Check solution to exam problem 4 on page 1

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

18.01

*Single Variable Calculus*, Fall 2006

Prof. David Jerison

**Course Material Related to This Topic:**

- Complete exam problem 7 on page 1
- Check solution to exam problem 7 on page 1

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

18.01

*Single Variable Calculus*, Fall 2006

Prof. David Jerison

**Course Material Related to This Topic:**

- Complete exam problem 8 on page 1
- Check solution to exam problem 8 on 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.

18.01

*Single Variable Calculus*, Fall 2006

Prof. David Jerison

**Course Material Related to This Topic:**

- Complete exam problem 6 on page 1
- Check solution to exam problem 6 on 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.

18.01

*Single Variable Calculus*, Fall 2006

Prof. David Jerison

**Course Material Related to This Topic:**

- Complete exam problem 5 to 6 on page 1
- Check solution to exam problem 5 to 6 on 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.

18.01

*Single Variable Calculus*, Fall 2006

Prof. David Jerison

**Course Material Related to This Topic:**

- Complete exam problem 8 on page 1
- Check solution to exam problem 8 on page 1

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.

18.01

*Single Variable Calculus*, Fall 2006

Prof. David Jerison

**Course Material Related to This Topic:**

- Complete exam problem 11 on page 2
- Check solution to exam problem 11 on page 1

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.

18.01

*Single Variable Calculus*, Fall 2006

Prof. David Jerison

**Course Material Related to This Topic:**

- Complete exam problem 14 on page 2
- Check solution to exam problem 14 on page 1