  # Calculus with Dr. Bob IV: Applied Integration

## Video Lectures

Displaying all 21 video lectures.
 Lecture 1 Play Video The Area between Two Curves 1Worked 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 Play Video The Area between Two Curves 2Worked 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 Play Video Example of Volume Using Cross Sectional AreaMultivariable 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 Play Video The Disk/Washer Method for Volume 1Worked 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 Play Video The Disk/Washer Method for Volume 2Worked 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 Play Video The Disk/Washer Method for Volume 3Worked 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 Play Video The Shell Method for Volume 1Worked 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 Play Video The Shell Method for Volume 2Worked 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 Play Video The Shell Method for Volume 3Worked 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 Play Video Formula for Arc Length 1Worked 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 Play Video Formula for Arc Length 2Worked 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 Play Video Formula for Arc Length 3Worked 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 Play Video Arc Length Along Parabola 1: Base CaseCalculus: 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 Play Video Arc Length Along Parabola 2: Sinh FormulaCalculus: 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 Play Video Arc Length Along Parabola 3: Log FormulaCalculus: 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 Play Video Area of a Surface of RevolutionWorked 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 Play Video Moments and Center of Mass 1 - Point Masses on a LineCalculus: 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 Play Video Moments and Center of Mass 2 - Point Masses in the PlaneCalculus: 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 Play Video Moments and Center of Mass 3 - Planar Lamina of Uniform DensityCalculus: 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 Play Video Moments and Center of Mass 4 - Integral Formula for Planar LaminaCalculus: We derive the integral formula for the moments of a planar lamina using the limit process for integration. Lecture 21 Play Video Moments and Center of Mass 5 - Rod of Nonuniform DensityCalculus: 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.