Double Integral Over General Region Calculator

What is a Double Integral Over General Region?

A double integral over a general region extends the concept of single integrals to two dimensions. It calculates the volume under a surface z = f(x,y) over a region D in the xy-plane defined by inequalities such as x ≥ 0, y ≥ 0, and x² + y² ≤ 1.

The general form is:

where D is the region of integration, and the limits of integration are defined by the region's boundaries.

How to Calculate a Double Integral Over General Region

1. Identify the region D in the xy-plane defined by inequalities.
2. Express the region as a type I or type II region based on the inequalities.
3. Set up the iterated integral with appropriate limits of integration.
4. Integrate with respect to the inner variable first, then the outer variable.
5. Evaluate the integral to find the volume under the surface.

The Formula

The double integral over a general region D is calculated using the iterated integral approach:

where:

• f(x,y) is the integrand function
• dA is the area element
• a and b are the outer limits of integration
• g₁(x) and g₂(x) are the inner limits of integration

Worked Example

Calculate ∫∫_D (x² + y²) dA where D is the region bounded by x = 0, x = 1, y = 0, and y = √(1 - x²).

1. Identify the region D: a quarter-circle of radius 1 in the first quadrant.
2. Set up the integral as a type I region:
   ∫₀¹ ∫₀^√(1-x²) (x² + y²) dy dx
3. Integrate with respect to y first:
   ∫₀¹ [x²y + (y³)/3] from 0 to √(1-x²) dx
4. Evaluate the inner integral:
   ∫₀¹ [x²√(1-x²) + (1-x⁴)/3] dx
5. Integrate with respect to x to find the final result.

The exact value of this integral is π/8.

Applications

Double integrals over general regions are used in:

• Calculating volumes under surfaces
• Finding areas of complex regions
• Computing probabilities in continuous probability distributions
• Physics applications involving density functions
• Engineering problems involving surface areas and masses