Expert ∫ Calculus Topic 88 of 104
Double Integrals in Polar
Use dA = r dr dθ.
Key idea
Use dA = r dr dθ.
In depth
For circular regions use x = r cos θ, y = r sin θ and dA = r dr dθ. The extra r accounts for area elements growing with distance from the origin.
Worked example
Area of a disk of radius a: ∫₀²π∫₀ᵃ r dr dθ = πa².
Real-world uses
- Finding the area of a circular sector or the mass of a disc with varying density
Common mistake
Forgetting the factor r.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Double Integrals in Polar?
Use dA = r dr dθ. It is a topic at the expert level in Calculus.
Where is Double Integrals in Polar used in real life?
Real-world examples: Finding the area of a circular sector or the mass of a disc with varying density.
What should I learn before Double Integrals in Polar?
Make sure you are comfortable with Double Integrals, the previous topic in Calculus.
What is a common mistake with Double Integrals in Polar?
Forgetting the factor r.
How can I practise Double Integrals in Polar?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Double Integrals in Polar?
Move on to Triple Integrals next.
Related expert topics in Calculus
- Derivatives of Inverse Trig – Derivatives of arcsin, arccos, arctan.
- Implicit Differentiation – Differentiate both sides of F(x,y)=0 treating y as a function of x.
- Logarithmic Differentiation – Take logs before differentiating products and powers.
- Related Rates – Link rates of change of connected quantities.
- Linear Approximation – Approximate f near a using its tangent line.
- Differentials – Use dy = f′(x)dx for small changes.