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Master Areas Between Curves: Calculus Techniques Explained

Demonstrate an understanding of the absolute value of real numbers

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Absolute value functions

Expressing an Absolute Value Linear Function as a Piecewise Function

Expressing an Absolute Value Quadratic Function as a Piecewise Function

Solving absolute value equations

Solving absolute value inequalities

Solving Absolute Value Inequalities Involving "less than"

Solving Absolute Value Inequalities Involving "greater than"

Recognizing Absolute Value Inequalities with "No Solution" or "All Real Numbers"

Solve problems involving operations on radicals and radical expressions with numerical and variable radicands

Conversion between entire radicals and mixed radicals

Adding and subtracting radicals

Combining Like Radicals

Adding and Subtracting Radicals

Multiplying radicals

Multiplying Radicals

Operations with radicals

Operations with Radicals

Rationalize the denominator

Solve problems involving radical equations limited to square roots

Solving radical equations

Solving Radical Equations Involving One Square Root Term

Solving Radical Equations Involving More than One Square Root Term

Determine equivalent forms of rational expressions

Partial fraction decomposition

Simplifying rational expressions and restrictions

Perform operations on rational expressions

Adding and subtracting rational expressions

Adding and Subtracting Rational Expressions

Multiplying rational expressions

Multiplying Rational Expressions

Multiplying Rational Expressions in Factored Form

Dividing rational expressions

Dividing Rational Expressions

Dividing Rational Expressions in Factored Form

Simplifying complex fractions

Solve problems involving rational equations

Solving rational equations

Applications of rational equations

Factor polynomial expressions

Factoring difference of cubes

Factoring sum of cubes

Factor by taking out the greatest common factor

Factor by taking out GCF

Factor by grouping

Factoring difference of squares: \(x^2 - y^2\)

Factoring difference of squares: <katex>x^2 - y^2</katex>

Factoring difference of squares: x^2 - y^2

Factoring trinomials

Factoring Trinomials

Graph and analyze absolute value functions to solve problems

Areas between curves

Analyze quadratic functions of the form y = a(x - p)2 + q

Transformations of quadratic functions

Converting from general to vertex form by completing the square

Shortcut: Vertex formula

Converting general form into vertex form by applying the vertex formula

Vertex Formula Application

Quadratic function in vertex form: y = a(x-p)^2 + q

Analyze quadratic functions of the form y = ax2 + bx + c

Characteristics of quadratic functions

Quadratic function in general form: y = ax^2 + bx + c

Graphing quadratic functions: General form VS. Vertex form

Finding the quadratic functions for given parabolas

Applications of quadratic functions

Solve problems involving quadratic equations

Solving quadratic equations by factoring

Solving quadratic equations by completing the square

Using quadratic formula to solve quadratic equations

Solving a quadratic equation with TWO REAL SOLUTIONS

Solving a quadratic equation with TWO COMPLEX SOLUTIONS

Nature of roots of quadratic equations: The discriminant

Applications of quadratic equations

Solve problems involving systems of linear-quadratic and quadratic-quadratic equations

System of linear-quadratic equations

System of quadratic-quadratic equations

Solve problems involving linear and quadratic inequalities in two variables

Graphing linear inequalities in two variables

Graphing quadratic inequalities in two variables

Graphing systems of linear inequalities

Solve problems involving quadratic inequalities in one variable

Solving quadratic inequalities

Analyze arithmetic sequences and series to solve problems

Arithmetic sequences

Arithmetic series

Arithmetic series formula

Sigma notation

Analyze geometric sequences and series to solve problems

Geometric sequences

Geometric series

Geometric Series

Infinite geometric series

Arithmetic mean vs. Geometric mean

Graph and analyze reciprocal functions

Graphing reciprocals of linear functions

Graphing reciprocals of quadratic functions

Demonstrate an understanding of angles in standard position 0° to 360°

Angle in standard position

Coterminal angles

Coterminal Angles

Reference angle

Solve problems using the three primary trigonometric ratios for angles from 0° to 360° in standard position

Unit circle

Find the exact value of trigonometric ratios

ASTC rule in trigonometry (All Students Take Calculus)

ASTC rule in trigonometry

Solve problems using the cosine law and sine law including the ambiguous case

Law of sines

Ambiguous case: SSA triangles

Law of cosines

Applications of the sine law and cosine law

Converting between degrees and radians

Trigonometric ratios of angles in radians

Pre-Calculus 11

Algebra and Number

Relations and Functions

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