18.01SC | Fall 2010 | Undergraduate

Single Variable Calculus

3. The Definite Integral and its Applications

Part B: Second Fundamental Theorem, Areas, Volumes

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The Second Fundamental Theorem of Calculus describes how integration is the opposite of differentiation. It allows us to understand functions defined in terms of definite integrals, some of which are impossible to describe using more elementary methods. This part of the course also covers the use of integration to calculate volumes of solids.

» Session 51: The Second Fundamental Theorem of Calculus
» Session 52: Proving the Fundamental Theorem of Calculus
» Session 53: New Functions From Old
» Session 54: The Second Fundamental Theorem and ln(x)
» Session 55: Creating New Functions Using the Second Fundamental Theorem
» Session 56: Geometric Interpretation of Definite Integrals
» Session 57: How to Calculate Volumes
» Session 58: Volume of a Sphere, Revolving About x-axis
» Session 59: Volume of a Parabaloid, Revolving About y-axis
» Problem Set 7

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Overview

In this session you will:

  • Do practice problems
  • Use the solutions to check your work

Problem Set

Use Applications of Integration (PDF) to do the problems below.

Section Topic Exercises
4B Volumes by slicing; Volumes of revolution 2 (Use parts 1e and 1g for the problem), 5
4C Volumes by shells 2, 3
4J Other applications 3

Solutions

Solutions to Applications of Integration problems (PDF)

This problem set is from exercises and solutions written by David Jerison and Arthur Mattuck.

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Overview

We can use definite integrals to create a new type of function – one in which the variable is the upper limit of integration! The second fundamental theorem of calculus tells us, roughly, that the derivative of such a function equals the integrand.

Lecture Video and Notes

Video Excerpts

Clip 1: Second Fundamental Theorem of Calculus

Clip 2: Using the Second Fundamental Theorem of Calculus

Recitation Video

Applying the Second Fundamental Theorem

Second Fundamental Theorem and Chain Rule

Second Fundamental Theorem and Quadratic Approximation

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Overview

In this session we use definite integrals to compute areas of regions between curves.

Lecture Video and Notes

Video Excerpts

Clip 1: Areas Between Curves

Clip 2: Example: Area Between x = y2 and y = x − 2

Recitation Video

Area Between the Graphs of Sine and Cosine

Area Between y=x3 and y=3x-2

Worked Example

Area of a Smile

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Overview

This session introduces the “method of shells” for calculating volumes. The chosen example also illustrates the importance of understanding the units used in a mathematical model.

Lecture Video and Notes

Video Excerpts

Clip 1: Volume of a Cauldron

Clip 2: Warning about Units

Recitation Video

Volume of a Paraboloid via Disks

Volume of Revolution via Shells

Worked Example

Volume of Revolution via Washers

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Course Info

Instructor
Departments
As Taught In
Fall 2010
Learning Resource Types
Exams with Solutions
Lecture Notes
Lecture Videos
Problem Sets with Solutions
Simulations
Recitation Videos