Preview Activity 12.2.1.
In this activity we will explore the double integral of \(f(x,y) = 25-x^2-y^2\) on the rectangular domain \(R = [-3,3] \times [-4,4]\text{.}\)
As with partial derivatives, we may treat one of the variables in \(f\) as constant and think of the resulting function as a function of a single variable. We will now investigate what this means geometrically and algebraically when we integrate with one input variable fixed.
(a)
Let \(a\) be a fixed value in the interval \([-3,3]\text{.}\) Compute
\begin{equation*}
\int_{-4}^4 f(a,y) dy
\end{equation*}
(b)
The answer of your previous task depended on the value of \(a\) but the process would be the exact same set of steps for every \(a\text{,}\) so we can define a function \(A(x)\) as
\begin{equation*}
A(x) = \int_{-4}^4 f(x,y) \, dy
\end{equation*}
Explain the geometric meaning of the value of \(A(x)\) relative to the surface defined by \(f\) in the context of the trace determined by the fixed value of \(x\text{.}\) Your explanation should refer to different parts of Figure 12.2.2
(c)
For a fixed value of \(x\text{,}\) say \(x_i^*\text{,}\) write a few sentences to explain the geometric meaning of \(A(x_i^*) \ \Delta x\text{.}\) Your explanation of \(A(x_i^*) \Delta x\) should refer to the parts of Figure 12.2.3.
(d)
Since \(f\) is continuous on \(R\text{,}\) we can define the function \(A = A(x)\) at every value of \(x\) in \([-3,3]\text{.}\) Now think about subdividing the \(x\)-interval \([-3,3]\) into \(m\) subintervals, and choosing a value \(x_i^*\) in each of those subintervals. What will be the meaning of the sum \(\sum_{i=1}^m A(x_i^*) \ \Delta x\text{?}\)
(e)
Explain why \(\int_{-3}^3 A(x) \, dx\) will determine the exact value of the volume under the surface \(z = f(x,y)\) over the rectangle \(R\text{.}\)
