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Master the Area of Polar Curves: Formulas & Techniques

Integral Calculus: Master Advanced Integration Techniques

Integrals

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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

Improper integrals

Improper Integrals

Integration by parts

Integration using trigonometric identities

Integration Applications

Areas between curves

Volumes of solids of revolution - Disc method

Volumes of solid with known cross-sections

Volumes of solids of revolution - Shell method

Volumes of solid of revolution - Shell method

Arc length

Average value of a function

Introduction to differential equations

Order and solutions to differential equations

Separable equations

Separable Equations

Applications to differential equations

Modeling with differential equations

Slope fields

Understanding Slope Fields

Euler's method

Euler's Method

Euler"s Method

Euler"s method

Sequence and Series

Introduction to sequences

Monotonic and bounded sequences

Introduction to infinite series

Convergence and divergence of normal infinite series

Convergence & divergence of geometric series

Divergence of Geometric Series

Convergence & divergence of telescoping series

Divergence of harmonic series

Divergence of Harmonic Series

P Series

Convergence and Divergence of P Series

Alternating series test

Alternating Series Test

Convergence of the Alternating Series Test

Divergence test

Divergence Test

Comparison & limit comparison test

Comparison & Limit Comparison Test

Integral test

Integral Test for Series Convergence

Convergence/Divergence of Integral Test

Ratio test

Ratio Test

Convergence & Divergence of Ratio Test

Root test

Root Test

Root Test for Series Convergence

Convergence & Divergence of Root test

Absolute & conditional convergence

Absolute & Conditional Convergence

Radius and interval of convergence with power series

Radius of Convergence of Logarithmic Function

Radius and Interval of Convergence with Power Series

Radius of Convergence of Tangent Function

Functions expressed as power series

Taylor series and Maclaurin series

Approximating functions with Taylor polynomials and error bounds

Parametric Equations and Polar Coordinates

Defining curves with parametric equations

Tangent and concavity of parametric equations

Area of parametric equations

Arc length and surface area of parametric equations

Arc Length of a Parametric Curve

Polar coordinates

Tangents of polar curves

Area of polar curves

Arc length of polar curves

Integral calculus made easy!
Get better marks in calculus class with our complete Integral Calculus help. Our calculus tutors cover all topics you will see in any typical Integration class that deals with single variable functions.

Keeping with your textbook and class, our Integral calculus video lessons walk you through all topics in integral calculus such as, Antiderivative, Integral of trig functions, Definite integral, Indefinite integral, Integration by substitution, Improper integrals and more. Learn the concepts with our tutorials that show you step-by-step solutions to even the hardest integral calculus questions. Then, reinforce your understanding with tons of integral calculus practice.
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Integral Calculus

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https://dmn92m25mtw4z.cloudfront.net/img_set/textbook-integral-calculus/v1/textbook-integral-calculus-168w.png

Intro

Intro to area formulas for single and overlapping polar curves

Ex.1

Finding the area of the inner loop of r = 1 + 2sin(θ)

Ex.2

Finding the area of the inner loop of r = 1 - 2cos(θ)

Ex.3

Area between two circles r=2 and r=3 using polar integration

Ex.4

Area between r=1 and r=1+sin(θ) using polar area formula

In this section, we will learn how to find the area of polar curves. For polar curves, we do not really find the area under the curve, but rather the area of where the angle covers in the curve. Note that not only can we find the area of one polar equation, but we can also find the area between two polar equations. It is important to always draw the curves out so that you can locate the area you are integrating, and write the integral correctly. We will first examine a generalized formula in finding areas of polar curves. Then we will apply this to finding areas with one polar curve and then two polar curves. Make sure you know your trigonometric identities very well before tackling these questions.

Area of Polar Curves: Mastering Calculations and Techniques

What You'll Learn

Apply the polar area formula A = ½(r²)dθ to find areas bounded by polar curves

Determine integration bounds (α and β) by finding curve intersections and analyzing geometry

Calculate areas of inner and outer loops by identifying where r = 0 and setting appropriate limits

Find areas between two polar curves using A = ½(r_outer² - r_inner²)dθ

Use trigonometric identities to simplify integrals involving sin²θ and cos²θ

What You'll Practice

Finding areas of inner loops where polar curves intersect themselves

Calculating enclosed areas between concentric circles and cardioid-like curves

Determining correct integration bounds by solving r = 0 and setting curves equal

Evaluating definite integrals with trigonometric substitutions and identities

Why This Matters

Before You Start — Make Sure You Can:

This Unit Includes

5 Video lessons

Practice exercises

Learning resources

Skills

Polar Coordinates

Definite Integrals

Trigonometric Identities

Curve Intersections

Area Calculations

Integration Bounds