Section 4.6 – Area Between Two Curves - Virtual Math Learning Center

Area of the Region Bounded by Two Curves

Finding the area of a region bounded by two curves using definite integrals

Calculating a Definite Integral From a Graph

Calculating a definite integral by finding the area under a curve using geometry

Definite Integral Comparison Properties

Comparing two definite integrals with the same limits of integration

Evaluating a Definite Integral

Using the Fundamental Theorem of Calculus to evaluate a definite integral

Evaluating a Definite Integral 2

Using the Fundamental Theorem of Calculus to evaluate a definite integral

Evaluating a Definite Integral with u-Substitution

Using the Fundamental Theorem of Calculus and \(u\)-substitution to evaluate a definite integral

Part 1 of the Fundamental Theorem of Calculus

Using Part 1 of the Fundamental Theorem of Calculus to find the derivative of a function defined using an integral

Solving a Word Problem With a Definite Integral

Solving a word problem using a definite integral and the Fundamental Theorem of Calculus

Average Value of a Function

Finding the average value of a a function on an interval using a definite integral

WIR 20B M152 V7

Finding the area of a region enclosed by two curves

WIR 20B M152 V8

Finding the area of a region enclosed by two curves

WIR 20B M152 V9

Finding the area of a region enclosed by two curves

WIR 20B M152 V10

Finding the area of a region bounded by curves

Area Between Curves

Using Riemann sums to find the definite integral for the area between curves

Integrating Along y-axis for Area Between Curves

Determining whether its easier to integrate along the \(x\) or \(y\)-axis to find the area between curves

Area Between Curves Using Multiple Integrals

Using multiple integrals to find the area between two curves

MATH 152: Area Exercise 1

Finding the area of a region bounded by two curves

MATH 152: Area Exercise 2

Finding the area of a region bounded by two curves

MATH 152: Area Exercise 3

Finding the area of a region bounded by two curves

MATH 152: Area Exercise 4

Finding an integral representing the area of a region bounded by two curves

MATH 152: Area Exercise 5

Finding the area of a region bounded by two curves

Area Between Curves

Using Riemann sums to find the definite integral for the area between curves

Integrating Along y-axis for Area Between Curves

Determining whether its easier to integrate along the \(x\) or \(y\)-axis to find the area between curves

Area Between Curves Using Multiple Integrals

Using multiple integrals to find the area between two curves

Area Between Curves: MATH 172 Problems 1-3

Interpreting integrals to represent areas between curves

Average Value of a Function: MATH 172 Problems 6-8

Calculating the average value of a function and proving the Mean Value Theorem for Integrals

Areas Between Curves in Python

Using Python to identify the region between curves and then evaluate the area of the region

Evaluating Limits from a Graph

Finding limits from the graph of a piecewise function

Evaluating One-Sided Limits from a Graph

Finding one-sided limits from the graph of a piecewise function

Evaluating a Limit Numerically

Finding the limit of a function numerically using a calculator

Combinations of Limits from Graphs

Finding limits of combinations of functions using their graphs

Evaluating Limits Algebraically

Using the properties of limits to find several limits algebraically

Limit of a Piecewise Function

Finding one and two-sided limits for a piecewise function algebraically

Limit of a Rational Function

Finding the limit of a rational function algebraically

Limit of a Rational Function 2

Finding the limit of a rational function algebraically

Limit with Multiple Fractions

Finding the limit algebraically of a function with multiple fractions

Limit of a Fraction with a Square Root

Finding the limit algebraically of a function with a square root in the numerator of a fraction

Limit with an Absolute Value

Finding the limit algebraically of a function with an absolute value

Limit at Negative Infinity of a Rational Function

Solving a limit approaching negative infinity for a rational function

Limits at Infinity of a Rational Function

Solving the limits approaching infinity and negative infinity of a rational function

Limits at Infinity of a Rational Function 2

Solving the limits approaching infinity and negative infinity of a rational function

Limits at Infinity of a Fraction with Exponential Functions

Solving the limits approaching infinity and negative infinity with exponential functions in the numerator and denominator

Vertical Asymptotes and Holes of a Rational Function

Finding the holes and vertical asymptotes of a rational function

Horizontal Asymptote of a Fraction with Exponential Functions

Finding the horizontal asymptotes of a function that is a fraction with exponentials in the numerator and denominator

Where the Graph of a Function is Not Continuous

Determining where a piecewise function is continuous from its graph

Continuity of a Function with a Cube Root

Finding where a function with a fraction and cube root is continuous

Continuity of a Function with a Square Root and Logarithm

Determining where a function with a fraction, square root, and logarithm is continuous

Continuity of a Piecewise Function

Determining where a piecewise function is continuous algebraically

Solving for a Constant so a Piecewise Function is Continuous

Solving for a constant to make a piecewise-defined function continuous for all real numbers

Average Rate of Change from a Table

Finding the average rate of change over time periods from a table

Slope of a Secant Line

Finding the slope of a secant line to a given function

Instantaneous Rate of Change of a Function

Finding the exact value of the instantaneous rate of change

Instantaneous Rate of Change of a Rational Function

Finding the exact value of the instantaneous rate of change

Instantaneous Rate of Change of Profit

Finding the instantaneous rate of change of profit for a given price-demand function and cost function

Slope of a Tangent Line

Finding the equation of the tangent line to a function using the limit definition

Finding the Derivative of a Polynomial Using the Limit Definition

Using the limit definition to find the derivative of a polynomial

Finding the Derivative of a Rational Function Using the Limit Definition

Using the limit definition to find the derivative of a rational function

Finding the Derivative of a Root Function Using the Limit Definition

Using the limit definition to find the derivative of a square root function

Determining Where the Derivative Exists From a Graph

Determining where the derivative of a function does not exist from a graph

Finding a Derivative with Exponential Functions and Logarithms

Finding the derivative a function with a power, exponential, and logarithm

Finding a Derivative by Simplifying a Rational

Finding the derivative of a rational function by simplifying and using the power rule

Evaluating Derivatives From a Given Table

Finding and evaluating derivatives using values given in a table

Calculating Marginal Revenue

Finding the marginal revenue given the price-demand function for a word problem

Using the Product Rule with a Logarithm

Finding a derivative of a function with a logarithm using the Product Rule

Using the Product Rule with an Exponential

Finding the derivative of a function with a square root and exponential using the Product Rule

The Quotient Rule with an Exponential

Finding the derivative of a fraction with an exponential using the Quotient Rule

Using Both the Quotient and Product Rule

Finding a derivative using both the Product and Quotient Rule

Using the Product and Quotient Rules with a Table

Finding a derivative using the Product and Quotient Rule and values given in a table

Using the Generalized Power Rule

Finding the derivative of a function using the Generalized Power Rule

Using the Generalized Exponential Rule

Using the Generalized Exponential Rule to find the derivative of a function

Using Log Properties to Simplify Before Taking the Derivative

Using logarithm properties to simplify before using the Chain Rule to find a derivative

Using Log Properties to Simplify Before Taking the Derivative 2

Using logarithm properties to simplify before using the Generalized Logarithm Rule to find a derivative

Using the Chain Rule with a Table

Using the chain rule to find derivates with some values given in a table

Calculating the Rate of Growth With Compound Interest

Finding the rate a balance is increasing for interest compounded monthly and continuously

Using the Alternate Form of the Chain Rule

Using the alternate form of the Chain Rule to find the derivative

Using Logarithmic Differentiation to Find a Derivative

Finding the derivative of a function that requires logarithmic differentiation

Using Implicit Differentiation to Find a Derivative

Using implicit differentiation to find the derivative of a function

Using Implicit Differentiation to Find a Derivative 2

Using implicit differentiation to find the derivative of a function

Solving a Related Rates Word Problem About Liquid Filling a Sphere

Finding the rate water is flowing into an expandable sphere given the rate a sphere is growing

Solving a Related Rates Word Problem About Profit

Finding the rate of change of profit given the rate of increase for sales

Solving a Related Rates Word Problem About a Sliding Ladder

Finding how fast a ladder moves up a wall as the bottom is being pushed

Finding Critical Values

Finding the critical values for several functions

Finding Where a Function is Increasing and Decreasing

Determining the intervals where a function is increasing and decreasing

Increasing and Decreasing and Local Extrema

Finding the intervals where a function is increasing/decreasing and the local extrema

Increasing and Decreasing and Local Extrema for a Revenue Function

Finding intervals of increase/decrease and the local extema for revenue from a word problem

Using the Quotient Rule Twice to Find the Second Derivative

Using the quotient rule twice to find the second derivative of a rational function

Inflection Points and Intervals of Concavity for an Exponential Function

Finding the intervals of concavity and inflection points for a function with exponentials

Inflection Points and Intervals of Concavity from a Graph

Determining the intervals of concavity and inflection points from the graph of a function

Local and Absolute Extrema from a Graph

Finding all local and absolute extrema of a function from a graph

Absolute Extrema on a Closed Interval

Finding the absolute maximum and minimum of a rational function on two closed intervals

Absolute Extrema on a Closed Interval 2

Finding the absolute maximum and absolute minimum of a rational function on two closed intervals

Maximizing a Product of Numbers

Finding the maximum product of two numbers given their sum is 100

Finding the Price to Maximize Profit

Finding the selling price of trucks to maximize profit given the cost and price-demand functions

Finding the Box with the Largest Volume

Finding the dimensions of the box with the largest volume given the amount of material to make the box

Minimizing the Cost of a Box

Finding the minimum cost of a box given the volume and cost of materials

Finding an Antiderivative

Finding the antiderivative of a function

Finding an Indefinite Integral

Evaluating an indefinite integral with an exponential

Finding an Indefinite Integral with a Fraction

Solving an indefinite integral by first simplifying the fraction

Finding a Function from its Derivative and One Value

Solving for a function given its derivate and one value

Finding the Cost Function from Marginal Cost

Solving the antiderivative of marginal cost to find the cost function and evaluating a particular cost

u-Substitution With an Exponential Function

Solving an indefinite integral with an exponential function suing u-substitution

u-Substitution With a Natural Logarithm

Solving an indefinite integral with a natural logarithm using u-substitution

u-Substitution With a Quotient and Exponentials

Solving an indefinite integral with exponentials in a quotient using u-substitution

u-Substitution With a Quotient and a Root

Solving an indefinite integral with a cube root in the denominator using u-substitution

Finding the Function from Its Derivative Using u-Substitution

Using u-substitution in an indefinite integral to find a function from its derivative and a given vaue

Calculating Profit From Marginal Profit Using u-Substitution

Calculating profit from marginal profit using an indefinite integral and u-substitution

Calculating the Exact Area Under a Curve With Geometry

Using geometric formulas for circles, triangles, and rectangles to find the exact area under a curve

Introduction to Riemann Sums

Estimating the area under a function using right-hand, left-hand, and midpoint Riemann sums

Estimating Area With a Right-Hand Riemann Sum

Estimating the area under a curve using right-hand Riemann sums

Estimating Area With a Left-Hand Riemann Sum

Estimating the area under a curve using left-hand Riemann sums

Estimating Area With a Midpoint Riemann Sum

Estimating the area under a curve using midpoint Riemann sums

Estimating the Area Under a Curve With a Riemann Sum

Estimating the area under a graph using right-hand Riemann sums

Properties of Definite Integrals

Evaluating a definite integral given the values of two other definite integrals of the same function

Riemann Sums and Definite Integrals: MATH 151 5.2 Problems 6-12

Using Reimann sums and the Fundamental Theorem of Calculus

Riemann Sums and Definite Integrals: MATH 151 5.3 Problems 6-12

Using Riemann sums and the Fundamental Theorem of Calculus

Integration by Substitution: MATH 151 5.5 Problems 13 & 14

Examples of integration by substitution

MLC WIR 20B M151 week9 #13

Using a limit of Riemann sums to write an expression for the area under a graph

MLC WIR 20B M151 week9 #14a

Evaluating a definite integral by using geometry to find the area under the curve

MLC WIR 20B M151 week9 #14b

Evaluating a definite integral by using geometry to find the area under the curve

MLC WIR 20B M151 week10 #21

Evaluating a definite integral by interpreting it as an area and using geometry

MLC WIR 20B M151 week10 #22

Using the Fundamental Theorem of Calculus to find the derivative of a function defined using an integral

MLC WIR 20B M151 week10 #24a

Evaluating a definite integral containing a polynomial

MLC WIR 20B M151 week10 #24b

Evaluating a definite integral by first simplifying the integrand

MLC WIR 20B M151 week10 #24c

Evaluating a definite integral with an exponential and sine function

MLC WIR 20B M151 week10 #24e

Using u-substitution to evaluate an indefinite integral

Review for the Common Exam: MATH 152 Exam 1 Review Problems 11-14

Review of work and average value

Review for the Common Exam: MATH 152 Exam 1 a Review Problems 1-4

Reviewing u-Substitution

Review for the Common Exam: MATH 152 Exam 1 b Review Problems 29-33

Review of trigonometric substitution

Review for the Common Exam: MATH 152 Exam 1 b Review Problems 34 & 35

Review of partial fractions

Review for the Common Exam: MATH 152 Exam 3 b Review Problems 4 & 5

Review of using tests to determine if a series is absolutely convergent, convergent, and divergent

Review for the Common Exam: MATH 152 Exam 3 c Review Problems 10-13

Review of power series properties and writing functions as power series

Review for the Common Exam: MATH 152 Exam 3 c Review Problems 13-18

Review of Taylor and Maclaurin Series and their properties

Review for the Common Exam: MATH 152 Exam 3 c Review Problems 4 & 5

Review of using tests to determine if a series is absolutely convergent, convergent, and divergent

WIR 20B M152 V88

Finding the length of a curve given by parametric equations

WIR 20B M152 V89

Find the integral representing the surface area of a rotated parametric curve

WIR 20B M152 V90

Finding the surface area of a rotated parametric curve

WIR 20B M152 V95

Finding the area of a region defined using polar coordinates

WIR 20B M152 V96

Finding the area of a region defined using polar coordinates

WIR 20B M152 V97

Finding the area of a region defined using polar coordinates

WIR 20B M152 V98

Finding the area of a region defined using polar coordinates

WIR 20B M152 V99

Finding the area of a region defined using polar coordinates

WIR 20B M152 V80

Evaluating a definite integral by rewriting the integrand as its Maclaurin series

Area with Riemann Sums and the Definite Integral

The definition of the Riemann Sum and how it relates to a definite integral

WIR 20B M152 V11

Finding the volume of a non-rotational solid

WIR 20B M152 V12

Finding the volume of a non-rotational solid

WIR 20B M152 V13

Finding the volume of a solid of revolution using the disk method

WIR 20B M152 V14

Finding the volume of a solid of revolution using the disk method

WIR 20B M152 V15

Finding the volume of a solid of revolution using the washer method

WIR 20B M152 V16

Finding the volume of a solid of revolution using the washer method

MATH 152: Volume Exercise 1

Using the disk method to find the volume of a rotational solid

MATH 152: Volume Exercise 3

Using the disk method to find the volume of a rotational solid

MATH 152: Volume Exercise 4a

Using the washer method to set up an integral for the volume of a rotational solid

MATH 152: Volume Exercise 4b

Using the washer method to set up an integral for the volume of a rotational solid

MATH 152: Volume Exercise 4cd

Using the washer method to set up an integral for the volume of a rotational solid

MATH 152: Volume Exercise 5

Finding the volume of a non-rotation solid

MATH 152: Volume Exercise 6

Finding the volume of a non-rotational solid

MATH 152: Volume Exercise 7

Finding the volume of a non-rotational solid

Deriving the Formula for Cylindrical Shells

Deriving the formula for the method of cylindrical shells

Cylindrical Shells when Revolving Around y-axis

Explaining how to use cylindrical shells when the region is rotated around the \(y\)-axis

MATH 152: Cylindrical Shells Exercise 1

Using cylindrical shells to find the volume of a region rotated around the \(y\)-axis

MATH 152: Cylindrical Shells Exercise 2

Using cylindrical shells to find the volume of a region rotated around the \(y\)-axis

MATH 152: Cylindrical Shells Exercise 7

Using disks and shells to find the volume of a rotational solid

WIR 20B M152 V17

Finding the volume of a solid of revolution using cylindrical shells

WIR 20B M152 V18

Finding the volume of a solid of revolution using cylindrical shells

WIR 20B M152 V19

Solving work and force problems for a spring

WIR 20B M152 V20

Finding the work to pump water out of a tank

WIR 20B M152 V21

Finding the work to pump water out of a tank

Work Done Stretching a Spring Using Hooke's Law

How to use Hooke's Law to calculate the work done in stretching a spring

Work Done Pumping Water Out of a Tank

Explaining how to use integration to calculate the work done pumping water out of a tank

MATH 152: Work Exercise 3

Calculating the work done moving an object with a variable force

MATH 152: Work Exercise 4

Calculating a spring constant and the work done stretching a spring

MATH 152: Work Exercise 5

Solving for a spring constant and calculating work done stretching a spring

MATH 152: Work Exercise 6

Calculating the work done pumping water out of a tank that is an upright cylinder

MATH 152: Work Exercise 7

Calculating the work done pumping water out of a rectangular tank

MATH 152: Work Exercise 8

Calculating the work done pumping water out of a tank shaped like a trough

MATH 152: Work Exercise 9

Calculating the work done pumping water out of a tank shaped like a trough

MATH 152: Work Exercise 10

Calculating the work done pumping water out of a cone-shaped tank

MATH 152: Work Exercise 11

Calculating the work done pulling a rope to the top of a building

MATH 152: Work Exercise 12

Calculating the work done pulling a rope and weight to the top of a building

MATH 152: Work Exercise 13

Calculating the work done pulling part of a rope to the top of a building

MATH 152: Integration by Parts Exercise 2

Solving a definite integral with an exponential function using integration by parts

MATH 152: Integration by Parts Exercise 7

Solving a definite integral with arctangent using integration by parts

MATH 152: Trigonometric Integrals Exercise 4

Using a trigonometric identity to integrate powers of sine and cosine

MATH 152: u-Substitution Exercise 2

Using u-substitution to evaluate a definite integral with a power of a polynomial in the denominator

MATH 152: u-Substitution Exercise 3

Using u-substitution to integrate a rational function

MATH 152: u-Substitution Exercise 4

Using u-substitution to evaluate a definite integral with a natural logarithm

MATH 152: u-Substitution Exercise 6

Using u-substitution on an indefinite integral with an exponential function, sine, and cosine

MATH 152: Partial Fraction Decomposition Exercise 2

Using partial fraction decomposition to rewrite and then integrate a rational function

MATH 152: Partial Fraction Decomposition Exercise 6

Using partial fraction decomposition to rewrite and then integrate a rational function

MATH 152: Trigonometric Substitution Exercise 5

Using trigonometric substitution to evaluate a definite integral

MATH 152: Trigonometric Substitution Exercise 6

Using trigonometric substitution to evaluate a definite integral

WIR 20B M152 V36

Determining if an improper integral converges or diverges

WIR 20B M152 V37

Determining if an improper integral converges or diverges

WIR 20B M152 V38

Determining if an improper integral converges or diverges

WIR 20B M152 V39

Determining if an improper integral converges or diverges

WIR 20B M152 V40

Using the comparison test to determine if an improper integral converges or diverges

WIR 20B M152 V41

Using the comparison test to determine if an improper integral converges or diverges

WIR 20B M152 V42

Using the comparison test to determine if an improper integral converges or diverges