Area by Double Integration Formula

TL;DR

This session explains how to find the area of enclosed regions in a plane using double integration.

Transcript

hello friends in this session we'll see and understand how to find the area by double integration so let us say that we are in a plane uh it could be cartesian coordinate or it could be polar coordinates so it could be like x y plane two dimension plane so we'll see both the coordinates one by one and let's start with the first coordinate that is c... Read More

Key Insights

• ✈️ Double integration is used to find the area of enclosed regions in a plane.
• 🐻‍❄️ The process differs in Cartesian and polar coordinates.
• ⛔ In Cartesian coordinates, the limits for integration depend on the functions involved.
• 🪈 The order of integration can be changed in Cartesian coordinates.
• 🐻‍❄️ In polar coordinates, the area is calculated by integrating r times dθ.
• 🐻‍❄️ The limits in polar coordinates are determined by the functions g(θ) and f(θ).
• 👻 Double integration allows for the accurate measurement of irregularly shaped regions.

Questions & Answers

Q: What is the purpose of double integration in finding the area of enclosed regions?

Double integration is used to account for both the width and height of small rectangular elements in the enclosed region, allowing us to find the overall area.

Q: How do we determine the limits for integration in Cartesian coordinates?

In Cartesian coordinates, the limits for integration depend on the functions involved. The lower limit for y is the function g(x), and the upper limit is the function f(x). The limits for x are x1 and x2.

Q: Can the order of integration be changed in Cartesian coordinates?

Yes, it is possible to change the order of integration in Cartesian coordinates. The limits for y would then be y1 and y2, while the limits for x would be g(y) and f(y).

Q: How is the area of an enclosed region calculated in polar coordinates?

In polar coordinates, the area is found by integrating r times dθ. The limits for r are g(θ) and f(θ), while the limits for θ are θ1 and θ2.

Summary & Key Takeaways

• The content discusses finding the area of an enclosed region in a plane using double integration.

• It explains the process for both Cartesian coordinates and polar coordinates.

• The double integration involves integrating with respect to x and y or theta and r, depending on the coordinate system.