Calculate The Double Integral 4xy 2 X 2 1da R

This guide explains how to calculate the double integral of the function 4xy over the region defined by 2 ≤ x ≤ 2 and 1 ≤ y ≤ a r. We'll cover the mathematical process, provide a working example, and discuss how to interpret the results.

What is a double integral?

A double integral extends the concept of single integration to two dimensions. It calculates the volume under a surface defined by a function f(x,y) over a region R in the xy-plane. The double integral is written as:

Double Integral Formula

∫∫_R f(x,y) dA = ∫_a^b ∫_c(x)^d(x) f(x,y) dy dx

For our specific problem, we're calculating the integral of 4xy over a region where x ranges from 2 to 2 and y ranges from 1 to a r. This means we're integrating over a line segment in the xy-plane, which is a special case of a double integral.

How to calculate the double integral

The process for calculating a double integral involves several steps:

Identify the function to be integrated (4xy in this case)
Determine the limits of integration for x and y
Integrate with respect to y first (inner integral)
Integrate the result with respect to x (outer integral)
Evaluate the definite integral using the given limits

Step-by-Step Calculation

1. Inner integral (with respect to y): ∫₁^{a r} 4xy dy

2. Solve the inner integral: 2x(a r)² - 2x(1)² = 2x(a r)² - 2x

3. Outer integral: ∫₂² [2x(a r)² - 2x] dx

4. Solve the outer integral: [(a r)²x² - x²] evaluated from 2 to 2

5. Final result: [(a r)²(2)² - (2)²] - [(a r)²(2)² - (2)²] = 0

This shows that when the limits for x are the same (2 to 2), the double integral evaluates to zero, which makes sense geometrically as we're integrating over a line segment.

Example calculation

Let's work through a concrete example with a = 3:

Example with a = 3

1. Inner integral: ∫₁³ 4xy dy = 2x(3)² - 2x(1)² = 18x - 2x = 16x

2. Outer integral: ∫₂² 16x dx = 8x² evaluated from 2 to 2 = 0

Final result: 0

This confirms our earlier result that the integral evaluates to zero when the x limits are equal.

Interpreting the result

The result of zero for this double integral has a specific geometric interpretation:

The function 4xy is being integrated over a line segment in the xy-plane
When integrating over a line segment, the volume under the surface is zero
This makes sense because a line has no area or volume in three dimensions

Key Insight

Double integrals over line segments always evaluate to zero because the region has no area in the plane of integration.

FAQ

Why does the double integral evaluate to zero when x limits are equal?

The integral evaluates to zero because we're integrating over a line segment, which has no area in the xy-plane. The volume under the surface over a line is zero.

What happens if I change the y limits?

If you change the y limits while keeping the x limits equal, the result will still be zero because the region remains a line segment with no area.

Can I calculate a double integral over a different region?

Yes, you can calculate double integrals over any region in the xy-plane. The process changes depending on whether the region is a rectangle, triangle, or other shape.

What's the difference between single and double integrals?

A single integral calculates area under a curve in one dimension, while a double integral calculates volume under a surface in two dimensions.