Calculus with Dr. Bob IV: Applied Integration

Video Lectures

Displaying all 21 video lectures.
Lecture 1
The Area between Two Curves 1
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The Area between Two Curves 1
Worked problem in calculus. The area between the curves g(x) = x^ - 3x + 2 and f(x) = 2-x is found by integrating both along the x and y axes.
Lecture 2
The Area between Two Curves 2
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The Area between Two Curves 2
Worked problem in calculus. The area between the curves y=x and y=x^3 - 8x is computed. The key step is to note that two integrals are needed as the top function changes at x=0.
Lecture 3
Example of Volume Using Cross Sectional Area
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Example of Volume Using Cross Sectional Area
Multivariable Calculus: We compute the volume of the solid with base equal to the disc with radius 7 and with cross-sections as equilateral triangles perpendicular to the base.
Lecture 4
The Disk/Washer Method for Volume 1
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The Disk/Washer Method for Volume 1
Worked problem in calculus. Let A be the area bounded by the curves f(x) = -x^2 + 3x + 1 and g(x) = x+1. The disk method is used to calculate the volume of the solid of revolution formed by revolving A about the (a) x-axis and (b) the line y=1.
Lecture 5
The Disk/Washer Method for Volume 2
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The Disk/Washer Method for Volume 2
Worked problem in calculus. Let A be the region bounded by y=x, y=2x, and y=4. The disk method is used to calculate the volume of the solid of revolution formed by revolving A about (a) the y-axis and (b) the line x= -2.
Lecture 6
The Disk/Washer Method for Volume 3
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The Disk/Washer Method for Volume 3
Worked problem in calculus. The disk method is used to verify the formulas for the volume of a sphere and the volume of a right circular cone.
Lecture 7
The Shell Method for Volume 1
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The Shell Method for Volume 1
Worked problem in calculus. The shell method is used to compute the volume of a sphere of radius r and a right circular cone with height h and base radius r.
Lecture 8
The Shell Method for Volume 2
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The Shell Method for Volume 2
Worked problem in calculus. Let A be the area bounded by the curves f(x) = x^2-2x and g(x) = 2x. The shell method is used to compute the volume generated by revolving A about (a) the y-axis and (b) the line x = -1.
Lecture 9
The Shell Method for Volume 3
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The Shell Method for Volume 3
Worked problem in calculus. Let A be the area bounded by the functions f(x) = x + 1, g(x) = 1, and h(x) = 2-x. The shell method is used to compute the volume of the solid of revolution generated by revolving A about (a) the x-axis and (b) the line y = 1.
Lecture 10
Formula for Arc Length 1
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Formula for Arc Length 1
Worked problem in calculus. The formula for the arc length of a curve is given and applied to (a) the line y=mx+b, (b) the upper semicircle of the unit circle, and (c) the catenary f(x) = (e^x + e^{-x})/2 (also known as hyperbolic cosine).
Lecture 11
Formula for Arc Length 2
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Formula for Arc Length 2
Worked problem in calculus. Using the Second Fundamental Theorem of Calculus, find the arc length of the curve along the function F(x) = int_1^x sqrt(t^4-1) dx above the x-axis from x=1 to x=2.
Lecture 12
Formula for Arc Length 3
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Formula for Arc Length 3
Worked problem in calculus. Given the arc length function for the graph of f(x) from 1 to x, how do we recover the original f(x)? We apply the Second Fundamental Theorem of Calculus to L(x) = 2/3 (x^{3/2}-1).
Lecture 13
Arc Length Along Parabola 1: Base Case
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Arc Length Along Parabola 1: Base Case
Calculus: We calculate the arc length along the graph of the parabola y = x^2 from x=0 to x=1. We use a hyperbolic trig substitution to find an antiderivative in the arc length function.
Lecture 14
Arc Length Along Parabola 2: Sinh Formula
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Arc Length Along Parabola 2: Sinh Formula
Calculus: Using the antiderivative in Part 1, we calculate the arc length along the parabola y = .1*x^2 - .1*x + 20 from x= 0 to x=20.
Lecture 15
Arc Length Along Parabola 3: Log Formula
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Arc Length Along Parabola 3: Log Formula
Calculus: We give the general formula for arc length along a parabola without using inverse sinh. We replace the hyperbolic sine with a better expression in the antiderivative of Parts 1 and 2 and provide a direct calculation of the formula using a regular trig substitution.
Lecture 16
Area of a Surface of Revolution
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Area of a Surface of Revolution
Worked problem in calculus. We derive the formula for the area of a surface of revolution and apply the formula to compute the surface areas of (a) a right circular cylinder (no bases), (b) a right circular cone (no base), and (c) a sphere of radius R.
Lecture 17
Moments and Center of Mass 1 - Point Masses on a Line
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Moments and Center of Mass 1 - Point Masses on a Line
Calculus: We define the moment of a point mass about a point P and extend the definition for a system of point masses about the origin. The center of mass is then defined and we explain the physical significance in terms of doors and see-saws.
Lecture 18
Moments and Center of Mass 2 - Point Masses in the Plane
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Moments and Center of Mass 2 - Point Masses in the Plane
Calculus: We consider moments and center of mass for a system of point masses in the plane. Three equations are given, and an example is computed.
Lecture 19
Moments and Center of Mass 3 - Planar Lamina of Uniform Density
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Moments and Center of Mass 3 - Planar Lamina of Uniform Density
Calculus: We present formulas for the moments and center of mass of a planar lamina of uniform density rho. Examples considered are (a) a rectangle of height h and base b, (b) the upper half of the unit disk, and (c) the region between y=x and y=x^2-x from x=0 to x=1. For part (c), we also show how to obtain the center of mass by splitting the region in two pieces.
Lecture 20
Moments and Center of Mass 4 - Integral Formula for Planar Lamina
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Moments and Center of Mass 4 - Integral Formula for Planar Lamina
Calculus: We derive the integral formula for the moments of a planar lamina using the limit process for integration.
Lecture 21
Moments and Center of Mass 5 - Rod of Nonuniform Density
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Moments and Center of Mass 5 - Rod of Nonuniform Density
Calculus: We derive the formula for the mass, moment about zero, and center of mass for a rod of nonuniform density rho(x). Examples are the rod from x=0 to x=L when rho(x) is constant and when rho(x) = x.