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Master Areas Between Curves: Calculus Techniques Explained

Business Calculus: Master Essential Math for Economics

Limits and Continuity

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Intermediate value theorem

Proving the Existence of Real Roots Using IVT

Intermediate Value Theorem

Application of Intermediate Value Theorem

Proving the Existence of a Real Root Using IVT

Finding limits from graphs

Continuity

Discontinuities of Rational Functions (denominator=0)

Finding limits algebraically - direct substitution

Finding limits algebraically - when direct substitution is not possible

Infinite limits - vertical asymptotes

Limits at infinity - horizontal asymptotes

Limit Evaluation

Limits at Infinity of Exponential Functions

Squeeze Theorem

Squeeze theorem

Quadratic Function with Integer Vertex

l'Hospital's Rule

l'Hopital's rule

l'Hospital's rule

Evaluating the limit using L'Hospital's Rule

l"Hospital"s rule

Limit Laws

Limit laws

Evaluating Limits of Functions

limit laws

Evaluating Limits with specific limits given

Introduction to Calculus - Limits

Derivatives

Estimating derivatives from a table

Chain rule

Chain Rule

Differentiation of Trigonometric Function

Differentiate Radical Functions with Chain Rule

Differentiate: Trigonometric Functions

Chain rule: Differentiate Trigonometric Functions

Differentiation of Exponential Function

Differentiation of Exponential Functions

Differentiation: Chain Rule with Logarithmic Functions

Differentiation of Logarithmic Function

Product rule

Quotient rule

Differentiation using Quotient Rule

Differentiation using Chain Rule and Quotient Rule

Derivative of inverse trigonometric functions

Definition of derivative

Definition of Derivative

Derivative using Definition

Vertical Tangent Line of a Function

Applications to definition of derivative

Power rule

Power Rule

Application of the Power Rule

Power rule application

Applying Power Rule

Power Rule with Negative Exponents

Power rule with rational exponents

Power rule and rational exponents

Derivative of trigonometric functions

Derivative of Trigonometric Functions

Derivative of Composite Trig Functions

Derivative of exponential functions

Derivative of Exponential Functions

Differentiation of Composite Functions

Derivative of power and exponential functions

Derivative of logarithmic functions

Derivative of log with arbitrary base

Derivative of Natural Logarithm Functions

Derivative of Logarithmic Functions

Implicit differentiation 

Higher order derivatives

Higher Order Derivatives

First and second derivatives

First and Second Derivatives

Second derivatives with implicit differentiation

Implicit Differentiation

Higher Order Derivatives with Repeating Patterns

Derivative Applications

Rolle's theorem

Quadratic approximation

Slope and equation of tangent line

Mean value theorem

Mean Value Theorem

Mean Value Theorem Application

Critical number & maximum and minimum values

Critical Number & Maximum and Minimum Values

Linear approximation

Use of Linear Approximations in Estimating Square Roots

Linearization of Radical Functions

Linearization of Polynomial Functions

Linearization of Rational Functions

Linearization of Exponential Functions

Linearization of Logarithmic Functions

Linear Approximation of Trigonometric Functions

Optimization

Rectilinear motion: derivative

Velocity of a Particle

Position velocity acceleration

Total Distance Traveled by a Particle

Comparative Analysis of Particle's Velocity and Acceleration

Curve sketching

Related rates

Business Derivative Application

Marginal profit, and maximizing profit & average profit

Elasticity of demand

Continuous compound interest

Continuous growth and decay

Demand, revenue, cost & profit

Marginal revenue, and maximizing revenue & average revenue

Marginal cost, and minimizing cost & average cost

Integrals

Antiderivatives

Derivatives and Antiderivatives

Derivative and Antiderivative of a Function

Riemann sum

Riemann Sum

Evaluating integrals with a Riemann sum

Evaluating integrals with a Riemann Sum

Riemann Sum with Trapezoid Approximation

Trapezoidal Approximation of Riemann Sums

Fundamental theorem of calculus

Fundamental Theorem of Calculus Part II

Fundamental Theorem of Calculus Part I

Fundamental Theorem of Calculus

Definite integral

Definite Integral

Integration Techniques

Numerical Integration

Numerical integration

U-Substitution

U-Substitution: Integrating Radical Functions

Not-So-Obvious U-Substitution

Trigonometric substitution

Trigonometric Substitution

Integration of rational functions by partial fractions

Integration by parts

Integration using trigonometric identities

Integral Applications

Area between curves

Areas between curves

Volume of solids of revolution - Disc method

Volumes of solids of revolution - Disc method

Volume of solids with know cross-sections

Volumes of solid with known cross-sections

Volume of solids of revolution - Shell method

Volumes of solid of revolution - Shell method

Average value of a function

Business Integral Application

Consumer and producer surplus

Continuous money flow

Business Calculus

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Ex. 1:

Ex. 2:

Ex. 3:

Intro

So we learn that we can find the area under the curve, but we can actually find the area between two curves by taking the difference between the top curve and bottom curve, and integrating it in terms of x! Just make sure to pick your lower and upper bound correctly so that you are actually finding the area you are looking for. However, there may be cases where you don't really know which is the top curve and which is the bottom curve. In this case, you would instead have a left curve and right curve. To find the area between, you would take the difference between the right curve and the left curve and integrate in terms of y. In this section, we will take a look at all of these cases and write the integral correctly.

Finding the area between curves