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Master Areas Between Curves: Calculus Techniques Explained

Explore the concepts of average and instantaneous rate of change

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Finding limits from graphs

Finding limits algebraically - direct substitution

Finding limits algebraically - when direct substitution is not possible

Difference quotient: applications of functions

Limit laws

Evaluating Limits of Functions

limit laws

Evaluating Limits with specific limits given

Introduction to Calculus - Limits

Use linearization to solve problems

Quadratic approximation

Solve problems involving related rates

Related rates

Determine the definite integral 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

Determine the antiderivative of a function

Antiderivatives

Derivatives and Antiderivatives

Derivative and Antiderivative of a Function

U-Substitution

U-Substitution: Integrating Radical Functions

Not-So-Obvious U-Substitution

Trigonometric substitution

Trigonometric Substitution

Integration by parts

Integration using trigonometric identities

Solve problems that involve the application of the integral of a function

Areas between curves

Volumes of solids of revolution - Disc method

Volumes of solid with known cross-sections

Volumes of solid of revolution - Shell method

Arc length

Average value of a function

Improper integrals

Improper Integrals

Determine the derivative of a function by applying the definition of derivative

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

Infinite limits - vertical asymptotes

Squeeze theorem

Quadratic Function with Integer Vertex

Definition of derivative

Definition of Derivative

Derivative using Definition

Vertical Tangent Line of a Function

Applications to definition of derivative

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

l'Hospital's rule

l'Hopital's rule

l'Hospital's Rule

Evaluating the limit using L'Hospital's Rule

l"Hospital"s rule

Apply derivative rules to determine the derivative of a function

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

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

Find derivatives of trigonometric 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

Apply the Chain Rule to determine the derivative of a function

Derivative of inverse trigonometric functions

Implicit differentiation

Solve problems involving inverse trigonometric functions

Finding inverse trigonometric function from its graph

Evaluating inverse trigonometric functions

Finding inverse reciprocal trigonometric function from its graph

Inverse reciprocal trigonometric function: finding the exact value

Inverse Reciprocal Trigonometric Function: Finding the Exact Value

Find limits and derivatives of exponential and logarithmic functions

Exponential growth and decay by a factor

Exponential decay: Half-life

Exponential growth and decay by percentage

Use calculus techniques to sketch the graph of a function

Slope and equation of tangent line

Limits at infinity - horizontal asymptotes

Limit Evaluation

Limits at Infinity of Exponential Functions

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

Position velocity acceleration

Velocity of a Particle

Total Distance Traveled by a Particle

Comparative Analysis of Particle's Velocity and Acceleration

Curve sketching

Use calculus techniques to solve optimization problems

Marginal profit, and maximizing profit & average profit

Optimization

Demand, revenue, cost & profit

Marginal revenue, and maximizing revenue & average revenue

Marginal cost, and minimizing cost & average cost

Calculus 120

math

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.

Area between curves

Finding the area between curves