Course Information

Course

Subject:	Mathematics
Topic:	Integral Calculus
Views:	39,308

Educator

Name:	Math Doctor Bob (Robert Donley)
Type:	Individual

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Calculus with Dr. Bob II: Basic Integration

Video Lectures

Displaying all 39 video lectures.

Lecture 1 Play Video	Definition of Antiderivative Calculus: We define an antiderivative of a function f and explain indefinite integration as a process that undoes differentiation. We show that all antiderivatives of a function may be obtained by finding a single antiderivative and adding constants. We emphasize checking work and note that correct answers may need more work due the constant of integration.
Lecture 2 Play Video	Antiderivative of a Polynomial Calculus: We note rules for the antiderivative of a sum and scalar multiple. With these, we are able to find the indefinite integral of any polynomial. We also give an example with negative powers of x and a cruder method for checking our answer.
Lecture 3 Play Video	Antiderivative of (x-1)(x-2)/sqrt(x^3) Worked problem in calculus. The antiderivative of (x-1)(x-2)/ sqrt(x^3) is calculated.
Lecture 4 Play Video	Basic Trig Antiderivatives Calculus: We note the identity int F' dx = F(x) + C and apply it to the basic trig functions. As examples, we compute int [cos(.5x) -sin(.5x)]^2 dx and int cos(x)/(1 - cos^2(x)) dx.
Lecture 5 Play Video	Antiderivative of tan^2(x) Worked problem in calculus. The antiderivative of tan^2(x) is computed.
Lecture 6 Play Video	Antiderivative of sin(x)/[1-sin^2(x)] Worked problem in calculus. The antiderivative of sin(x)/[1-sin^2(x)] is calculated.
Lecture 7 Play Video	Visualizing an Antiderivative Calculus: We explain the connection from the graph of a function f to its antiderivatives F + C. We consider the special case of f(x) = 2x + 2 and draw an analogy to the point-slope description of a line.
Lecture 8 Play Video	Graphing f(x) from f'(x) Worked problem in calculus. The graph of the derivative f'(x) is given. We show how the graph of f(x) is obtained.
Lecture 9 Play Video	Antiderivative of a Piecewise-Defined Function Calculus: Suppose f(x) is continuous on the real line, f(0) = 10, f(10) = 100, and f'(x) = x+1 on x lt 0, r on 0 lt x lt 20, and 5 on x gt 20. Find f(x) and compare its graph with f'(x).
Lecture 10 Play Video	Solving Differential Equations Worked problem in calculus. We solve the differential equation f"(x) = x^{-1/2} + 1, f'(9) = 7, and f(4) = 32/3.
Lecture 11 Play Video	Equations of Motion Worked problem in calculus. From a height of 50m, an object is tossed into the air. If the maximum height is 150m, what was the initial velocity?
Lecture 12 Play Video	Overview of Rectangular Approximation of Area Calculus: We describe a process for approximating the area under the graph of a positive function using rectangles. We apply this process to the special case of f(x) = 1 + x over the interval [1, 2] with 10 and n rectangles. As a final note, we define sigma notation, partitions, and Riemann sums.
Lecture 13 Play Video	Rectangular Approximation of Area Worked problem in calculus. The area under the graph f(x) = sqrt(4-x^2) over [0,2] is approximated using 6 rectangles (upper and lower).
Lecture 14 Play Video	Overview of Summation Formulas Calculus: We review some basic summation formulas and rules for calculating Riemann sums. Over the range i from 1 to n, we state formulas for the sum of a constant r, i, i^2 and i^3. We review sigma notation and prove the rules for r, i and i^2. We also work out the special case of sum 1 to 4 of (i+1)^2 using our formulas.
Lecture 15 Play Video	Limit Summation Formula Worked problem in calculus. We compute a closed formula for the summation over i (from 1 to n) of (1 + i/n)^2 (1/n). Then the limit is taken as n goes to infinity.
Lecture 16 Play Video	General Method for Integer Power Sum Formula Calculus: We give a general method for deriving the closed formula for sums of powers of 1 through N. The technique uses the partial sum formula for geometric power series.
Lecture 17 Play Video	Riemann Sum with Signed Area Calculus: Calculate the Riemann sum for the function f(x) = (x-1)^2-1 over the interval [1,3] with 10 rectangles and using right endpoint. The graph is below the x-axis on [1,2]. We also compute the Riemann sum for n rectangles and let n go to infinity.
Lecture 18 Play Video	Limit Process for Area Worked problem in calculus. The limit process for area is carried out for f(x) = 4x - x^3 over the interval [0,2].
Lecture 19 Play Video	Definition of Definite Integral Calculus: We review the process for finding the area under a curve and give a definition for a definite integral when f(x) is continuous over an interval [a,b]. We state basic rules for definite integrals. Examples include f(x) = x+2 over [-1, 1] and f(x) = -sqrt(4-x^2) over [-2,2].
Lecture 20 Play Video	Definite Integral as Area 1 - Using the Graph Worked problem in Calculus. The definite integral int_0^2 sqrt(4x-x^2) dx is computed by graphing the curve and applying a geometric formula for the area.
Lecture 21 Play Video	Definite Integral as Area 2 - Breaking Up the Region Worked problem in calculus. Given the graph of f(x), we compute definite integrals of f(x) using geometric methods.
Lecture 22 Play Video	Definite Integral as Area 3 - Area Below the x-axis Worked problem in calculus. Find a and b such that the definite integral int_a^b -6+5x-x^2 dx is maximized.
Lecture 23 Play Video	The First Fundamental Theorem of Calculus Calculus: We state and prove the First Fundamental Theorem of Calculus. Before the proof, we show the following examples: (a) constant function (rectangle), (b) line through origin (triangle), (c) f(x) = sin(x) over [0, pi], and (d) f(x) = (x-1)^2 -1 over [1,3]. The key step in the proof is to reinterpret the Mean Value Theorem. We also note the interpretation for motion in a line.
Lecture 24 Play Video	Area Under a Curve 1 Worked problem in calculus. The area under the curve f(x) = \|x^2 - 4x +3\| over the interval [0, 4] is calculated using the First Fundamental Theorem of Calculus. The key step is break the interval into three regions.
Lecture 25 Play Video	Area Under a Curve 2 Worked problem in calculus. We calculate the area bounded by the curve f(x) = x^3 - x and the x-axis using the First Fundamental Theorem of Calculus.
Lecture 26 Play Video	The Mean Value Theorem for Integrals Calculus: We state and prove the Mean Value Theorem for Integrals. Examples include (a) f(x) = x+ 2 over the interval [1, 3], and (b) f(x) = x^5 - x over the interval [0, 1].
Lecture 27 Play Video	Example of Mean Value Theorem for Integration Worked problem in calculus. An example of the Mean Value Theorem is worked out for f(x) = sin(x) over the interval [0, pi/2].
Lecture 28 Play Video	The Second Fundamental Theorem of Calculus Calculus: We state and prove the Second Fundamental Theorem of Calculus. A formula is given for an antiderivative of f(x) when continuous on [a,b]. We interpret the result in terms of the area function for f(x) and note some examples and applications.
Lecture 29 Play Video	Example of 2nd Fundamental Theorem of Calculus 1 Worked problem in calculus. The Second Fundamental Theorem of Calculus is used to graph the area function for f(x) when only the graph of f(x) is given.
Lecture 30 Play Video	Example of 2nd Fundamental Theorem of Calculus 2 Worked problem in calculus. The Second Fundamental Theorem of Calculus is combined with the chain rule to find the derivative of F(x) = int_{x^2}^{x^3} sin(t^2) dt.
Lecture 31 Play Video	Integration By Substitution: Antiderivatives Calculus: We introduce the technique of integration by substitution for indefinite integrals. This is an integration method that reverses the chain rule for derivatives. Examples include (a) int x(1+x^2)^3 dx, (b) int x^6 cos(2x^7) dx, (c) int sin(x)cos(x) dx, and (d) int x(1+x)^2 dx. In problem (c), we recall an occasional problem with integrals, and, in problem (d), we show an instance where re-substitution is required.
Lecture 32 Play Video	Integration by Substitution: Definite Integrals Calculus: We note how integration by substitution works with definite integrals using the First Fundamental Theorem of Calculus. Two methods are given, and examples used are (a) int_0^1 x(x^2+1)^5 dx, (b) int_0^{pi/4} tan(x) sec^2(x) dx, (c) int_{-1}^1 x(1-x^2)^2 dx, and (d) int_0^2 x(1-x^2)^2 dx.
Lecture 33 Play Video	Example of Integration by Substitution 1: f(x) = (-x)/[(x+1)-sqrt(x+1)] Worked problem in calculus using integration by substitution. Calculate the indefinite integral of (-x) / [(x+1) - sqrt(x+1)]. Steps include a difference of two squares and checking the answer.
Lecture 34 Play Video	Example of Integration by Substitution 2: f(x) = x^5/(1-x^3)^3 Worked problem in calculus. Find the indefinite integral of x^5/[(1-x^3)^3]. Methods include integration by substitution and a re-substitution when some terms remain.
Lecture 35 Play Video	Example of Integration by Substitution 3: f(x) = x^2(1+x)^4 over [0,1] Worked problem in calculus. Compute the definite integral of x^2(1+x)^4 from x=0 to x=1. Integration by substitution gives a useful trick for when the 4 is replaced by higher exponents.
Lecture 36 Play Video	Example of Integration by Substitution 4: f(x) = (2x+3)/sqrt(2x+1) Calculus. Compute the indefinite integral of (2x+3)/sqrt(2x+1). Methods include integration by substitution and re-substitution.
Lecture 37 Play Video	Example of Integration by Substitution 5: f(x) = sin(x)/cos^3(x) Calculus: Find the indefinite integral of sin(x)/[cos^3(x)]. Integration by substitution is used. A second approach allows us to use the easily overlooked form [f]^1 f' in the substitution.
Lecture 38 Play Video	Example of Trapezoid Rule with Error Bound Calculus: The Trapezoid Rule is used to approximate the area under the curve f(x) = (1+x)^2 over the interval [0,2]. A bound for the error in the approximation is also given.
Lecture 39 Play Video	Example of Simpson's Rule with Error Bound Calculus: Simpson's Rule is used to approximate the area under the curve f(x) = sqrt(1 + x) over the interval [0, 1]. An upper bound for the error is also given.