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5.5 The Substitution Rule Substitution Rule Definite Integral with Substitution, U-Substitution, U-substitution example 2, U-substitution Example 3, Another u-substitution example, U-substitution and back substitution, Indefinite Integration (part IV) Integration using U-Substitution, Integration by U-Substitution, Definite Integral, U-substitution - Indefinite Integral, Another 2 examples, U-Substitution - More Complicated Examples Definite Integral by Substitution | MIT 18.01SC Single Variable Calculus, Fall 2010 Section 5.5 Indefinite Integration by U Substitution, Definite Integration with U Substitution Integration by Substitution how to integrate using u substitution How To Integrate Using U-Substitution, U-substitution With Definite Integrals Math 2B. Calculus. Lecture 06. The Substitution Rule., Differential & Integral Calculus, Lec 21, Math 31A, UCLA
6.1 Area Between Curves Introduction to Areas Between Curves, Basic example on finding the area between two curves, Another example on finding the area between two curves, Finding Area Between Curves by Integrating With Respect To y Area between curves Finding Areas Between Curves, Area Between Curves - Integrating with Respect to y Area between curves Section 6.1 Area Between 2 Curves using Vertical and Horizontal Representative Rectangles Finding Area Between Two Curves Area Between Two Curves Math 2B. Calculus. Lecture 07. Areas Between Curves
6.2 Volume Solid of Revolution (part 1), Solid of Revolution (part 2), Solid of Revolution (part 3), Solid of Revolution (part 4), Solid of Revolution (part 5), Solid of Revolution (part 6), Solid of Revolution (part 7), Solid of Revolution (part 8) Volumes of Revolution - Disk/Washers Example 1 Lecture 22 Section 6.2 Volume of Solid of Revolution Disk Method and Washer Method Volume of Solids By Disks and Washers Method Disk & Washer Method Math 2B. Calculus. Lecture 08. Calculating The Volume of Solids.
6.3 Shell method for rotating around horizontal line Longer Version - Volumes using Cylindrical Shells, Volumes of Revolution - Cylindrical Shells Volume of Revolution via Shells | MIT 18.01SC Single Variable Calculus, Fall 2010 Section 6.3 Volume of Solid of Revolution Shell Method 3 Examples Calculus 1 Lecture 5.3: Volume of Solids By Cylindrical Shells Method Shell Example 1, Shell Example 2, Shell Example 3, Shell Example 4 Shell Method - Volume of Revolution
6.4 Work in the case of constant force, Finding work done by springs, Example on finding the work done when stretching a spring, Example on finding the work done to lift a rope, Finding the work done in lifting problems Calculating the Work Required to Drain a Tank, Finding Work using Calculus - The Cable/Rope Problem, Finding Work using Calculus - The Cable/Rope Problem - Part b, Section 6.4 Work Done by a Variable Force Work Example 1, Work Example 2, Work Example 3, Work Example 4, Work Example 5 Work Problems, Work Done By a Variable Force Physics Problems, Force Displacement Graphs
7.1 Integration by Parts Introduction to Integration by Parts, Two examples using integration by parts, Two more examples using integration by parts, Volume application using integration by parts Deriving integration by parts formula, Integration by parts of (e^x)(cos x), Integration by parts twice for antiderivative of (x^2)(e^x), Antiderivative of xcosx using integration by parts Integration by Parts, Integration By Parts - Using IBP's Twice, Integration by Parts - Definite Integral, Integration by Parts - A Loopy Example! Lec 30 | MIT 18.01 Single Variable Calculus, Fall 2007, Finding u and v' When Integrating by Parts | MIT 18.01SC Single Variable Calculus, Fall 2010 Section 7.1 Integration by Parts 6 Examples LIATE, Definite Integration by Parts, Integration by Parts 2 Times 2 Examples Integration by Parts Integration by Parts Integration By Parts, 3- Product Term, Tablular Method Math 2B. Calculus. Lecture 10. Integration by Parts
7.2 Trigonometric Integrals Trig Antiderivative Rules and Key Trig Identities, Integrating products of powers of sines and cosines, Example on integrating products of even powers of sines and cosines, Example on integrating products of powers of tangents and secants, Another example on integrating products of powers of secants and tangents, Example on integrating powers of tangent Trig identities for trig integral Trigonometric Integrals - Part 1 of 6, Trigonometric Integrals - Part 2 of 6, Trigonometric Integrals - Part 3 of 6, Trigonometric Integrals - Part 4 of 6, Trigonometric Integrals - Part 5 of 6, Trigonometric Integrals - Part 6 of 6 Lec 27 | MIT 18.01 Single Variable Calculus, Fall 2007, Trig Integral Practice | MIT 18.01SC Single Variable Calculus, Fall 2010, Trig Integrals and a Volume of Revolution | MIT 18.01SC Single Variable Calculus, Fall 2010 Section 7.2 Trigonometric Integrals Trigonometric Integrals Example 1, Example 2, Example 3, Example 4, Example 5, Example 6 Trigonometric Integrals Math 2B. Calculus. Lecture 11. Trigonometric Integrals
7.3 Trigonometric Substitution Introduction to Trig Substitution, Example using trig sub on an indefinite integral , Example requiring completing the square before using trig substitution, Another example requiring completing the square before using trig substitution, Example using trig sub on a definite integral Integrals: Trig Substitution 1, Integrals: Trig Substitution 2, Integrals: Trig Substitution 3 (long problem), Trig and U substitution together (part 1), Trig and U substitution together (part 2) Trigonometric Substitution - Example 1, Trigonometric Substitution - Example 2, Trigonometric Substitution - Example 3 / Part 1, Trigonometric Substitution - Example 3 / Part 2, Trigonometric Substitution, Ex 4 - Rational Powers, Trigonometric Substitutions - More Examples, Trigonometric Substitution and a Definite Integral , Trigonometric Substitution Example 1 Part 1, Trigonometric Substitution Example 1 Part 2 Lec 28 | MIT 18.01 Single Variable Calculus, Fall 2007 Section 7.3 Trigonometric Substitution 1, Tirognometric Substitution 2 Trigonometric Substitution Example 1, Example 2, Example 3, Example 4, Example 5 Trigonometric Substitution Math 2B. Calculus. Lecture 12. Trigonometric Substitution
7.4 Integration of Rational Functions Partial Fraction Decomposition, Introduction, Forms, Example 1, Example 2, Example 3, Example 4, Example 5, Example 6 Partial fraction expansion to evaluate integral Partial Fraction Decompositions, Integration Using method of Partial Fractions , Integration Using Partial Fractions / Rationalizing Sub MIT Lecture 29 Section 7.4 Partial Fractions 1, Partial Fractions 2, Partial Fractions 3, Integration by Partial Fractions Example 1, Example 2, Example 3, Example 4, Example 5 Integration By Partial Fractions, Partial Fraction Decomposition Math 2B. Calculus. Lecture 13. Integration by Partial Fractions
7.6 Section 7.6 Integration by Tables Integration using Tables
7.7 Approximate Integration Trapezoidal approximation of area under curve, Rectangular and trapezoidal Riemann approximations Simpsons Rule - Approximate Integration, Simpson's Rule - Error Bound, Simpson's Rule - The Derivation, The Trapezoid Rule for Approximating Integrals Using the Trapezoid and Simpson's rules | MIT 18.01SC Single Variable Calculus, Fall 2010, Explanation of Simpson's rule | MIT 18.01SC Single Variable Calculus, Fall 2010 Section 7.7 Trapezoidal Rule Numerical Integration With Trapezoidal and Simpson's Rule Midpoint Rule vs. Trapezoidal Rule, Error Bounds, Trapezoidal Rule Example Trapezoidal Rule, Trapezoidal Rule & Simpson's Rule
7.8 Improper Integrals Introduction to Improper Integrals, Examples of improper integrals with infinite intervals, Evaluating an improper integral of type 1 using multiple integration techniques, Definition of Improper Integrals of Type II and three examples, Example evaluating an improper integral using integration by parts, Introduction to Comparison Theorem for Improper Integrals and an Example, Examples using the Comparison Theorem for Improper Integrals Introduction to improper integrals, Improper integral with two infinite bounds, Divergent improper integral Improper Integral - Basic Idea and Example, Improper Integral - More Complicated Example, Improper Integral - Infinity in Upper and Lower Limits, Improper Integral w/ Infinite Discontinuity in the Middle, Improper Integral with Infinite Discontinuity at Endpoint Improper Integrals | MIT 18.01SC Single Variable Calculus, Fall 2010, Lec 36 | MIT 18.01 Single Variable Calculus, Fall 2007 Section 7.8 Improper Integrals Improper Integrals Improper Integral Example 1, Improper Integral Example 2, Comparison Theorem for Improper Integral, Comparison Theorem Example Improper Integrals Math 2B. Calculus. Lecture 16. Improper Integrals
8.1 Arc Length Introduction to arc length, First example using arc length formula, Another example using arc length formula, Definition of arc length function and an example Non rigorous justification for arc length formula, Arc length integration example, Another arc length integration example Arc Length Formula - Example 1, Arc Length, Arc Length Formula - Example 2 Section 8.1 Arc Length of a Curve ( Smooth Curve ) 5 Examples Finding the Length of a Curve on a Plane Proof, Example Arc Length Problems Math 2B. Calculus. Lecture 17. Arc Length, Review Integration Techniques
8.2 Introduction to Surface Area , Examples on finding the surface area when a curve is rotated about the x-axis, Examples finding surface area when a curve is rotated about the y-axis Finding Surface Area - Part 1, Finding Surface Area - Part 2 Section 8.2 Area of a Surface of Revolution 1, Area of a Surface of Revolution 2, Proof, Example Surface Area
8.3 How to compute hydrostatic force on a dam, Example computing the hydrostatic force on a vertical plate Hydrostatic Force - Complete Example #1, Hydrostatic Force - Basic Idea / Deriving the Formula, Hydrostatic Force - Complete Example #2, Part 1 of 2, Hydrostatic Force - Complete Example #2, Part 2 of 2, Centroid Part 1, Centroid Part 2 Section 8.3 Fluid Pressure and Fluid Force Force Example 1, Force Example 2, Force Example 3, Force Example 4, Force Example 5, Force Example 6, Centroid Proof, Centroid Example Hydrostatic Force Problems, Center of Mass and Centroid Problems, Moment, Center of Mass, and Centroid
9.1 Modelling with Differential Equations Modeling population with simple differential equation The Logistic Equation and Models for Population - Example 1, part 1 Section 9.1 Verifying Particular Solutions to Differential Equations
9.3 Separable Equations Separable differential equations introduction | First order differential equations Solving Separable First Order Differential Equations - Ex 1, Separable Differential Equation, Example 2 Separable Equations Section 9.3 Intro to Solving Separable Differential Equation, Separable Differential Equations & Growth and Decay Model Solving First Order Differential Equations by Separation of Videos Separable Equation Example, Separable Equation Application Separable First Order Differential Equations
9.5 Linear Equations First Order Linear Differential Equations Solutions of First-order Linear Equations Section 9.5 Intro to First-Order Equations, Solving Linear Differential Equations with an Integrating Factor First Order Linear Differential Equation & Integrating Factor First Order Linear Differential Equations
10.1 Introduction to Curves Defined by Parametric Equations, Four examples on sketching parametric curves and eliminating the parameter, Creating parametric equations for a given Cartesian equation Parametric Equations 1, Parametric Equations 2 Parametric Equations - Some basic questions, Parametric Curves - Basic Graphing Parametric equations, arclength, surface area Section 10.1 Introduction to Parametric Equations Parametric Equations Parametric Equations Example Parametric Equations Introduction, Eliminating The Paremeter t, Graphing Plane Curves
10.2 Introduction to the parametric form of the derivative, Finding horizontal and vertical tangents to parametric curves, Finding the area of a region enclosed by a parametric curve, Finding the arc length of a parametric curve, Finding the surface area when a parametric curve is rotated about the x or y axis Derivatives of Parametric Fuctions Parametric equations, arclength, surface area Section 10.2 Calculus of Parametric Equations Arc Length Example Derivatives of Parametric Functions, Tangent Lines of Parametric Curves
10.3 Introduction to Polar Curves, Example on converting a polar equation to a Cartesian equation, Basic Polar Curves, Sketching polar curves Polar Coordinates 1 Polar Coordinates - The Basics Polar coordinates; area in polar coordinates Section 10.3 Understanding Polar Coordinates Using Polar Coordinates and Polar Equations Polar Review 1, , Polar Slope 1, Polar Slope 2, Polar Slope 3 Polar Coordinates Introduction, Graphing Polar Curves, Tangent Line Equations, Slope, & Derivatives In Polar Form
10.4 Area and Lengths in Polar Coordinates I, Area and Lengths in Polar Coordinates II, Area and Lengths in Polar Coordinates III, Area and Lengths in Polar Coordinates IV, Area and Lengths in Polar Coordinates V Justification for polar arc length formula Finding Areas in Polar Coordinates Polar coordinates; area in polar coordinates Section 10.4 Calculus of Polar Equations Area Example 1, Area Example 2 Finding Area In Polar Coordinates, Finding Area Bounded By Two Polar Curves, Arc Length of Polar Curves
10.5 Review of Conic Sections Introduction to Conic Sections Section 10.5 Finding Equations of Conics from Given Conditions Circle and Parabola, Ellipse and Hyperbola Parabola, Ellipse, Hyperbola Conic Sections, Parabolas, Hyperbolas, Ellipses Introduction to Conic Sections
10.6 Polar Form of a Conic: Finding the Type, the Directrix and Eccentricity Section 10.6 Polar Equations of Conic Sections In Polar Coordinates Conic Sections -- Polar Coordinate System
11.1 Sequences Introduction to sequences and series, Introduction to Taylor Polynomials, Definition and Examples of Geometric Sequences Sequences and Series (part 1), Convergent and divergent sequences Infinite series and convergence tests Section 11.1 Convergence and Divergence of Sequences Example 1, Example 2, Example 3, Example 4, Example 5, Example 6, Example 7, Example 8, Example 9, Example 10 Converging and Diverging Sequences Using Limits, Monotonic Sequences and Bounded Sequences Sequences, Math 2B. Calculus. Lecture 19. Sequences
11.2 Infinite Series Introduction to Infinite Series, Definition of series convergence and two examples, Example on determining convergence or divergence of a telescoping series, Geometric Series Test and Several Examples Divergence Test (nth Term test), Geometric Series Geometric Series and the Test for Divergence - Part 1, Geometric Series and the Test for Divergence - Part 2, Test for Divergence for Series, Two Examples p-Series Test Section 11.2 p-Series Test, Geometric and Telescoping Series Series, Geometric Series, Harmonic Series, and Divergence Test Divergence Test Example Convergence and Divergence - Introduction to Series, Harmonic Series, Telescoping Series, Geometric Series and Geometric Sequences - Basic Introduction, Divergence Test Infinite series of numbers, Math 2B. Calculus. Lecture 20. Series
11.3 The Integral Test Introduction to the Integral Test, Example on applying the Integral Test, Another example applying the Integral Test, p-Series Test and Examples, Introduction to Estimating the Sum of a Series Integral test intuition , Integral test to show series divergence Integral Test - Basic Idea, Integral Test for Series Example 1, Integral Test for Series - Example 2, Integral Test to Evaluate Series, Ex 3, Using the Integral Test for Series Integral Test, Integral Test as Estimation Section 11.3 Integral Test Integral Test for Convergence/Divergence for Series Integral Test Rationale and Examples, Integral Test Example Integral Test For Convergence and Divergence of Series, Remainder Estimate For The Integral Test The Integral Test, Math 2B. Calculus. Lecture 21. Integral Test and Estimates of Sums
11.4 Comparison Tests Introduction to Comparison Test and Examples, Discussion of Limit Comparison Test and an Example, Three examples applying the Limit Comparison Test Comparison Test, Limit Comparison Test Limit Comparison Test and Direct Comparison Test, Limit Comparison Test and Direct Comparison Test, Direct Comparison Test / Limit Comparison Test for Series - Basic Info, Limit Comparison Test for Series - Another Example 1, Limit Comparison Test for Series - Another Example 2 Comparison Tests Section 11.4 Limit Comparison Test, Direct Comparison Test Comparison Tests Limit Comparison Test, Example Direct Comparison Test, Limit Comparison Test Comparison Test, Math 2B. Calculus. Lecture 22. The Comparison Test, Math 2B. Calculus. Lecture 23. The Comparison Test/ Alternating Series Test, Part II
11.5 Alternating Series Introduction to Alternating Series, Introduction to Alternating Series Test, Four examples on applying the Alternating Series Test, Alternating Series Estimation Theorem Alternating series test, Alternating series error estimation Alternating Series, More Alternating Series Examples, Alternating Series - Another Example 2, Alternating Series Estimation Theorem Section 11.5 Alternating Series Test Alternating Series Test Alternating Series Test, Alternate Series Estimation Theorem Math 2B. Calculus. Lecture 23. The Comparison Test/ Alternating Series Test, Part II
11.6 Absolute Convergence and the Ratio/Root Tests Introduction to Absolute and Conditional Convergence and 3 Examples, Example on using absolute convergence to determine convergence of a series, Introduction to the Ratio Test Two examples applying the Ratio Test, Example on determining absolute or conditional convergence or divergence, Introduction to the Root Test and an Example Absolute Convergence, Conditional Convergence and Divergence Ratio Test Example Section 11.6 Example 1, Example 2 Root and Ratio Test Ratio Test Example 1, Ratio Test Example 2, Ratio Test Example 3, Ratio Test Example 4, Root Test Example 1, Root Test Example 2 Absolute Convergence, Conditional Convergence, and Divergence, Ratio Test, Root Test
11.8 Power Series Introduction to Power Series, Three basic examples on finding where a power series converges, Theorem on the 3 possibilities for the set of x for which a power series converges, Example on determining radius and interval of convergence of a power series Power series function representation using algebra Power Series Representation of Functions Power Series Practice Section 11.8 Interval of Convergence of Power Series Power Series Radius and Interval of Convergence, Example 1, Example 2 Power Series - Finding The Radius & Interval of Convergence Math 2B. Calculus. Lecture 25. Power Series
11.9 Representing Functions as Power Series Introduction to Representations of Functions as Power Series, Three examples on manipulating series and differentiating series, Three examples on integrating series, Approximating a definite integral using series and an estimation theorem Power series representation using integration Power Series Representation of Functions Section 11.9 Power Series Representation of Functions, Differentiation and Integration of a Power Series Power Series Example 1, Example 2, Example 3, Example 4, Example 5, Example 6 Power Series - Representation of Functions, Power Series - Differentiation and Integration Math 2B. Calculus. Lecture 26. Representing Functions as Power Series Representing Functions as Power Series
11.10 Taylor and Maclaurin Series Introduction to Taylor and Maclaurin Series, Two examples on finding Taylor and Maclaurin series, Discussion of Taylor's Inequality and when a function equals its Taylor Series, Two examples on proving a function equals its Taylor series, Additional examples on working with Taylor Series Maclaurin and Taylor Series Intuition, Taylor Polynomials Taylor and Maclaurin Series - Example 1, Finding a Taylor Polynomial to Approximate a Function Taylor Series, Taylor's Series of a Polynomial, Finding Taylor Series Section 11.10 Taylor Series and Maclaurin Series Taylor Series and Maclaurin Series Taylor Series, Example 1, Example 2, Example 3 Taylor Series and Maclaurin Series Math 2B. Calculus. Lecture 27. Taylor Series and Maclaurin Series
11.11 Approximating functions using Taylor polynomials, Example on approximating a function at a particular value using Taylor polynomials Taylor Error 1, Taylor Error 2 Taylor's Inequality, Example 1, Example 2, Example 3, Example 4 Section 11.11 Taylor Polynomials and Estimation Error Approximation of Functions by Taylor Polynomials Taylor Polynomials & Maclaurin Polynomials With Approximations, Taylor's Remainder Theorem