Calculating the Volume Under a Surface Using Double Integration: A Comprehensive Guide

When dealing with complex shapes and surfaces in calculus, integrating over a specific region in the two-dimensional plane is often required. This article will guide you through the process of calculating the volume of a space above a square region and below the surface defined by the equation z √x √y. We will use double integration, a powerful tool in calculus for this purpose.

Understanding the Problem

The problem at hand involves a square region R bounded by the lines x 0, x 1, y 0, and y 1 in the xy-plane. We need to calculate the volume of the space above this region and below the surface defined by z √x √y. This can be mathematically expressed as:

Breaking Down the Integration

Since the function z √x √y is symmetric, we can simplify the problem by considering only the region where x≥0 and y≥0, and then multiplying the result by 2. Thus, we have:

Graphical Interpretation

For a better understanding, let’s visualize the problem. Consider two scenarios to gauge the volume.

Case 1: A Flat Plane

Imagine a flat plane defined by z 2. The area in the xy-plane is a square with side length 1, so its area is 1. The volume under this plane is the product of the area and the height above the plane, which is 2. Hence, the total volume is 2.

Case 2: The Curved Surface

Now, let’s consider the surface given by z √x √y. By comparing the volumes, we see that the volume under this surface is definitely less than 2, as the surface is curved. Graphing the surface helps visualize the shape of the volume we are trying to calculate.

Double Integration Approach

Using the method of double integration, we can systematically integrate the function over the square region. Each small area element dA dx dy contributes to the volume through the height z √x √y.

The volume at any point (x, y) is given by:

The total volume is the sum of all these small volume elements:

Mathematical Steps

To calculate the volume, we set up the double integral as follows:

Evaluating the inner integral:

Conclusion

The volume of the space above the square region and below the surface defined by z √x √y is given by V 4/9. This value represents the precise volume under the surface, which is less than the volume of the flat plane scenario previously considered.

By following the method of double integration, we can accurately calculate volumes under complex surfaces, providing a strong foundation for further studies in calculus and related fields.