91Ó°ÊÓ

A surface integral is a way to extend the concept of integration to surfaces. Unlike a regular integral, which moves over a line or area, a surface integral spans a two-dimensional surface. Imagine draping a sheet over some terrain; the surface integral calculates how that sheet behaves, factoring in the curves and slopes of the land beneath.

In the context of the problem, the surface integral helps us find the total area of a surface over a specific region. This surface is defined in the 3D space, determined by the given equation of the surface and the corresponding boundaries on the xy-plane. Here, we utilize the formula:
\[\iint_R \sqrt{1 + \left(\frac{\partial f}{\partial x}\right)^2 + \left(\frac{\partial f}{\partial y}\right)^2} \, dx \, dy\]
This formula takes into account the two-dimensional "bumpiness" or "height differences" of the surface by incorporating the partial derivatives of the surface function. Calculating the surface integral involves determining these partial derivatives and applying them over the designated region "R."

A double integral is an extension of a regular integral that calculates over an area in a plane, typically the xy-plane. When using a double integral, we consider not just the accumulation in a single direction but across an entire plane. This is particularly useful for calculating volumes beneath a surface or areas over a region.

In this exercise, we set up the double integral to determine the surface area over the triangular region stated in the problem. We defined limits for both x and y based on the triangle's bounds. Specifically:

• For x: From 0 to 2
• For y: From 0 to 3x

These bounds define the area over which the double integral operates, gathering all the infinitesimally small contributions of area from within the triangular region. Through evaluating the double integral:
\[\int_{0}^{2} \int_{0}^{3x} \sqrt{x^2 + 2} \, dy \, dx\]
We find the total surface area by accumulating these contributions over the entire specified region.

Partial derivatives are a foundational concept in multivariable calculus. They measure how a multi-variable function changes as one of its variables changes, while the others remain constant. It’s like determining the slope of a mountain path when you only adjust your elevation or your north-south position without moving east-west.

In surface area calculations, partial derivatives provide the necessary rates of change along each primary axis in the xy-plane. They refine our understanding of the surface's slope and how much more "area" that slope adds. For our surface defined by the equation \(z = \frac{x^2}{2} - y\), we find the partial derivatives:

• With respect to x: \(\frac{\partial f}{\partial x} = x\)
• With respect to y: \(\frac{\partial f}{\partial y} = -1\)

These derivatives are then used in the surface integral formula to accurately account for the surface’s curvature over the triangular region.

The substitution method is a classical technique in calculus for simplifying the evaluation of integrals. It involves substituting an expression with another variable, making the integral easier to solve. This is akin to changing the units in a problem to ones that make arithmetic straightforward.

In the final steps of integration in this problem, we used substitution to simplify the integral
\[\int 3x\sqrt{x^2 + 2} \, dx\]
After letting \(u = x^2 + 2\), we derived \(du = 2x \, dx\), allowing the integral to be rewritten in terms of \(u\). This change streamlined the problem, freeing us from complex terms in x and turning it into:
\[\frac{3}{2} \int \sqrt{u} \, du\]
By replacing the bounds to accommodate the new variable, the integral could be easily evaluated. This useful trick reaffirmed the power of substitution in tackling otherwise complicated integrals by shifting to simpler forms.