Approximating a Solid by Rectangular Prisms

About integration for a function defined on a region

If a solid can be modeled by an integrable function of two variables over a general region in the plane, we can use integration to find the volume of the solid. Let’s consider a region \(D\) on the \(x-y\) plane, which is bounded by two vertical lines and two functions of \(x\). The two vertical lines are \(x=a\) and \(x=b\) respectively, and the two functions of \(x\) is \(y=u(x)\) and \(y=v(x)\). The integrable function is \(f(x,y)\). The integration to find the volume of the solid can be calculated as following: \[V=\int^b_a{\int^{v(x)}_{u(x)}{f(x,y)dy}dx}\] The double integral \(\int^{b}_{a}{\int^{v(x)}_{u(x)}{f(x,y)dy}dx}\) of positive \(f(x,y)\) can be regarded as the volume under the surface \(z=f(x,y)\) over the region \(D\). We can use the Riemann sum to approximate the integral: \[\sum_{i,j}{f(x_i,y_j)}\Delta x \Delta y \] Each term in the Riemann sum is the volume of a thin box with base \(\Delta x \times \Delta y \) and height \( f(x_i,y_j)\).

The solid to be analyzed

The function we chose is a hyperbolic paraboloid. The surface defined by the hyperbolic paraboloid is also called saddle, as it is visually similar to a saddle for horse-riding. The hyperbolic paraboloid function is defined by \[z=f(x,y)=1+2x^2-2y^2\]

The region on the \(x-y\) plane is bounded by two vertical lines and two functions of \(x\). The two vertical lines are \(x=-0.6\) and \(x=0.6\) respectively, and the two functions \(y=u(x)\) and \(y=v(x)\) are hyperbolas: \[ y=u(x)=-\sqrt{0.28+x^2}, -0.6 \le x \le 0.6\] \[ y=v(x)=\sqrt{0.28+x^2}, -0.6 \le x \le 0.6\]

The solid can be defined by the saddle surface and the bounded region as the following picture shows:

This solid was chosen for the following reasons:
1. The function of the saddle surface has a concise form, so that we can focus on the concept of approximating the volume by double integral. The saddle surface we chose is not only important from the mathematical point of view, but also popular such as a saddle for horse riding in our daily lives. The Gaussian curvature, which is the product of the two principal curvatures \(Κ = κ_1κ_2\), is negative for a saddle shaped surface.
2. The region on the \(x-y \) plane is not a simple rectangle, but a somewhat irregular region bounded by two vertical lines and hyperbolas. By using numerical approximation, i.e. the Riemann sum, we can easily calculate the volume of the solid over the region.
The functions we chose made a shape with irregular surface and base region. This helps us to understand the concept and the calculation process of using the double integral to find the volume of the solid.

Compute the volume of approximation

We can compute the volume of solid approximated by rectangular prisms. The calculation process is shown in the following table

From the table, the volume of approximation \(V'\) is 1.488 cubic inches.
The 3D shape of the approximation is shown in the following picture:

If more rectangle prisms are used, it is possible to get more accurate approximation.

Compute the actual volume

The actual volume can be calculated by double integration: \[V=\int^{-0.6}_{0.6}{\int^{\sqrt{0.28+x^2}}_{-\sqrt{0.28+x^2}}{(1+2x^2-2y^2)dy}dx}=1.493\] The actual volume of the solid is 1.493 cubic inches. The difference between the approximated volume and the accurate volume is \( \Delta V= |V-V'|=1.493-1.488=0.005\) cubic inches. The relative error is \( \frac{\Delta V}{V}= 0.005/1.493 ≈ 0.33\% \).
From the above example, we can better understand the concept and the numerical approximation of the double integral of a function over a region on a plane.