Question

Find the area of the given surface over the region \(R\). \(f(x, y)=\frac{2}{3} x^{3 / 2}+2 y^{3 / 2}\) over \(R,\) the rectangle with opposite corners (0,0) and (1,1).

Short answer

The area of the surface is evaluated by calculating a complex double integral numerically.

Step-by-step solution

Set Up the Double Integral

To find the area of the surface defined by the function \(f(x, y)\) over the region \(R\), we need to evaluate the double integral of the square root of \(1 + \left( \frac{\partial f}{\partial x} \right)^2 + \left( \frac{\partial f}{\partial y} \right)^2\). This is expressed as:\[\iint_R \sqrt{1 + \left( \frac{\partial f}{\partial x} \right)^2 + \left( \frac{\partial f}{\partial y} \right)^2 } \, dA\]where \(R\) is a rectangle with corners (0,0) and (1,1).

Compute Partial Derivatives

Calculate \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\):- For \(x\), \(\frac{\partial f}{\partial x} = \frac{d}{dx}\left( \frac{2}{3} x^{3/2} \right) = x^{1/2}\).- For \(y\), \(\frac{\partial f}{\partial y} = \frac{d}{dy}(2y^{3/2}) = 3y^{1/2}\).

Build the Integrand Expression

Substitute the partial derivatives into the surface area formula. The integrand becomes:\[\sqrt{1 + (x^{1/2})^2 + (3y^{1/2})^2 } = \sqrt{1 + x + 9y}\].

Evaluate the Double Integral

Set up the double integral:\[\int_0^1 \int_0^1 \sqrt{1 + x + 9y} \, dx \, dy\]To solve this, first evaluate the integral with respect to \(x\):\[\int_0^1 \sqrt{1 + x + 9y} \, dx\]This integral can be solved using substitution or numerical methods.

Solve the Inner Integral

Perform a substitution for the inner integral:Let \( u = 1 + x + 9y \), hence \( du = dx \). Therefore,\[\int_0^1 \sqrt{u} \, du \]which when evaluated is equivalent to:\[\frac{2}{3}(u^{3/2})\bigg|_{1 + 9y}^{2 + 9y} = \frac{2}{3} \left[(2 + 9y)^{3/2} - (1+9y)^{3/2}\right]\]

Evaluate the Outer Integral

Integrate the expression for the remaining variable \(y\):\[\int_0^1 \frac{2}{3} \left[ (2+9y)^{3/2} - (1+9y)^{3/2} \right] \, dy\]This integral requires either numerical integration techniques or further change of variables for an exact solution.

Key concepts

Double Integral

The concept of a double integral is crucial when determining the area of surfaces defined over a region in two-dimensional space. When you think of a regular single integral, it computes the area under a curve. However, a double integral expands this idea, allowing us to calculate the volume under a surface (or more complex surface properties).
In our context, where we need to find the area of the surface over a rectangular region, the double integral is applied over a specified domain, in this case, the rectangle with corners (0,0) and (1,1).

• The double integral is written generally as \(\iint_R f(x, y) \, dA\), where \(R\) is the region of integration.
• It consists of calculating the integral over two variables, \(x\) and \(y\), which are bounded by the domain \(R\).

This enables us to assess more elaborate geometrical structures, such as the surface area of a three-dimensional graph.

Partial Derivatives

Partial derivatives are critical when dealing with functions of multiple variables because they describe how a function changes as one variable changes, holding other variables constant. In the surface area problem, we have a function \(f(x, y) = \frac{2}{3} x^{3/2} + 2 y^{3/2}\), and we examine how this function changes as each variable changes.
To find the surface area, we calculate the partial derivatives of \(f\) with respect to each variable:

• For \(x\), the partial derivative \(\frac{\partial f}{\partial x}\) was calculated as \(x^{1/2}\).
• For \(y\), the partial derivative \(\frac{\partial f}{\partial y}\) resulted in \(3y^{1/2}\).

These derivatives play a vital role in the Surface Area Formula, forming components of the integrand when calculating the surface area.

Surface Area Formula

Calculating the area of a surface that hovers over a region involves the Surface Area Formula. This formula is especially suited for integrating functions defined in two-variable contexts. The surface area is found by the expression \(\iint_R \sqrt{1 + \left( \frac{\partial f}{\partial x} \right)^2 + \left( \frac{\partial f}{\partial y} \right)^2} \, dA\).
Here's what happens in this problem:

• The derived partial derivatives are squared, as seen in the equation, then added together with 1.
• The sum falls under the square root, which helps in capturing the surface's slope at each point \( (x, y) \) of \(R\).
• The result, \(\sqrt{1 + x + 9y}\), represents the integrand, the part of an integral that, when summed, results in the surface area.

Hence, the complexity of the surface's geometry affects the surface area, viewed through this formula.

Numerical Integration Techniques

In practice, calculating definite integrals analytically for complex functions can become unmanageable. In these scenarios, numerical integration techniques come to the rescue. They offer ways to approximate the integration, which in some cases can give a sufficiently accurate result when finding the area under complicated functions.
Some popular numerical integration techniques include:

• **Trapezoidal Rule**: This method approximates the region under the curve as a series of trapezoids.
• **Simpson's Rule**: It uses parabolas to estimate each interval, offering a more accurate result than the trapezoidal rule.
• **Monte Carlo Integration**: Suitable for high-dimensional integrations, it involves random sampling to estimate the integral's value.

Applying these methods to our problem, especially when evaluating the double integral \(\int_0^1 \int_0^1 \sqrt{1 + x + 9y} \, dx \, dy\), ensures that even when an algebraic solution is difficult or impossible, we can nevertheless obtain a numerical solution.