Question

In Exercises 7 and \(8,\) use a computer algebra system to evaluate \(\int_{S} x y d S .\) $$ S: z=9-x^{2}, \quad 0 \leq x \leq 2, \quad 0 \leq y \leq x $$

Short answer

The double integral for a computer algebra system to evaluate would be: \[ \int_{0}^{2} \int_{0}^{x} x y d y d x \]

Step-by-step solution

Define the function and regions

Firstly, we need to define the function that has to be evaluated and the regions over which it would be evaluated. The function to be evaluated is \(f(x,y)=xy\) and the region S is limited by the inequalities: \(0 \leq y \leq x \leq 2\). This implies we are integrating over a triangular region in the \(xy\)-plane.

Express in terms of double integral

Next, we must express our surface integral as a double integral over the projection of the surface \(S\) onto the plane. For the surface integral on \(S\) of \(f(x, y)\), it can be expressed as a double integral over the region \(D\) of the projection of \(S\) onto the \(xy\)-plane, where \(D\) is derived from the given inequalities. Thus, \[ \int_{S} x y d S = \int \int_{D} x y dA \]

Set up the limits for 'x' and 'y'

Now, we should set up the limits for both 'x' and 'y' according to the given limits. By inspecting the region restrictions, the limits for 'x' are \[0, 2\] and for 'y' are \[0, x\]. Consequently, the integral becomes: \[ \int_{0}^{2} \int_{0}^{x} x y d y d x\].

Compute the integral

At the end, you would use a computer algebra system to evaluate this double integral.