Activity 12.1.2.
In this activity, we will be going through the first two steps of the classic calculus approach to defining the definite integral of \(f(x,y) = 100 - x^2-y^2\) on the rectangular domain \(R = [0,8] \times [2,6]\text{.}\) Remember that we are trying to measure the volume below the graph of \(f\) over the region \(R\text{,}\) so we will start with estimating this volume.
(a)
We want to understand the numerical calculations involved in the classic calculus approach, so we will not look a graph of \(z=f(x,y)\text{,}\) but we will need to understand our region of integration. On Figure 12.1.7, outline \(R\text{,}\) the rectangle that corresponds to the region of integration.
(b)
Because all of the regions and subregions we are looking at will be rectangles, we can break up the \(x\) and \(y\) coordinates into pieces separately. We will break the interval of \(x\)-coordinates into four equally sized subintervals and we will break the interval of \(y\)-coordinates into three equally sized subintervals. How large will the subintervals be in the \(x\)-direction and the \(y\)-direction?
(c)
Let \(\Delta x_i\) be the \(i\)-th subinterval for \(x\text{.}\) We want to state the endpoints of each of the \(\Delta x_i\text{.}\) The first subinterval, \(\Delta x_1\text{,}\) will go from \(x_0\) to \(x_1\text{,}\) the second subinterval, \(\Delta x_2\text{,}\) will go from \(x_1\) to \(x_2\text{,}\) the third subinterval, \(\Delta x_3\text{,}\) will go from \(x_2\) to \(x_3\text{,}\) and the fourth subinterval, \(\Delta x_4\text{,}\) will go from \(x_3\) to \(x_4\text{.}\)
Give the values for \(x_0\text{,}\) \(x_1\text{,}\) \(x_2\text{,}\) \(x_3\text{,}\) and \(x_4\) and add these as tick marks on the \(x\)-axis of Figure 12.1.7 to make sure your subintervals are equally sized.
(d)
Let \(\Delta y_i\) be the \(i\)-th subinterval for \(y\text{.}\) Give the values for \(y_0\text{,}\) \(y_1\text{,}\) \(y_2\text{,}\) and \(y_3\) and add these as tick marks on the \(y\)-axis of Figure 12.1.7 to make sure your subintervals are equally sized.
(e)
Now that we have our subintervals in \(x\) and \(y\) coordinates, we want to state the smaller rectangles we will use for our approximation. Let \(R_{i j}\) be the rectangle corresponding to \(\Delta x_i \times \Delta y_j\text{.}\)
-
How many smaller rectangles are there in our partition?
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Outline each of the smaller rectangles on Figure 12.1.7 and label each rectangle as either \(R_{1 1}, R_{1 2} , ...\text{.}\)
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Let \(\Delta A\) be the area of each of these smaller rectangles (\(R_{i j}\)). \(\Delta A = \)
(f)
We will estimate the height of each of the rectangular prisms over our \(R_{i j}\) by evaluating \(f\) at the upper right point of each smaller rectangle. State the point at the upper right of each smaller rectangle and evaluate \(f\) at each of these points.
(g)
Write a sentence about why the volume of our approximation on each smaller rectangle would be
\begin{equation*}
f(x_i,y_j) \Delta A
\end{equation*}
then write a couple sentences about how you would find your approximation of the volume under the surface \(z=f(x,y)\) over the region \(R\text{.}\) (You do not need to do this calculation but rather explain what calculation is being done.)
(h)
We used the upper right point to approximate the heights of our rectangular prisms on each \(R_{i j}\text{.}\) Write a sentence or two about whether you think the upper right point provides an overestimate, an underestimate, or approximately the average value for \(f\) on each \(R_{i j}\text{.}\) Explain how this suggests that your estimate for the volume under the surface \(z=f(x,y)\) over the region \(R\) is either an overestimate, an underestimate, or approximately the correct value.
