Active Calculus - Multivariable

Subsection 12.1.1 Introduction

   

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Preview Activity 12.1.1.

(a)

A plot of \(f\) for inputs in the interval \([0,2]\) is shown in Figure 12.1.1. Break the interval \([0,2]\) into four equally sized subintervals and draw the rectangles that would be used to construct a Riemann sum to approximate the area under \(f\) on the interval \([0,2]\text{.}\) You can use whichever point you want on each subinterval to evaluate the height the of the rectangles.

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(b)

Estimate the heights of the rectangles used in your Riemann sum above to estimate \(\displaystyle{\int_0^2 f(x)\enspace dx}\text{.}\) Write a couple of sentences about why you think your estimate for the definite integral of \(f\) is either an overestimate, an underestimate, or close to the true value.

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(c)

Explain why it doesn’t matter what method (left endpoint, right endpoint, midpoint, etc.) you use for selecting which point is evaluated on each of the subintervals in the definition of the definite integral.

\begin{equation*} \int_a^b f(x) \enspace dx = \lim_{\Delta x_i \to 0} \sum_{i=1}^n f(x_i^*)\Delta x_i \end{equation*}

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(d)

For each of the following functions plotted below, determine if the definite integral over the region shown will be positive, negative, or zero and write a sentence to justify your answer.

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Subsection 12.1.2 Double Riemann Sums over Rectangles

   

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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.

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(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.

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(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?

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(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 \(x_4\text{.}\)

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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.

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(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.

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(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 \(\Delta x_i \times \Delta y_j\text{.}\)

1. How many smaller rectangles are there in our partition?
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2. 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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3. Let \(\Delta A\) be the area of each of these smaller rectangles (\(R_{i j}\)). \(\Delta A = \)
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(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.

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(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.)

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(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.

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Subsection 12.1.3 Double Riemann Sums and Double Integrals

Subsection 12.1.4 Interpretation of Double Riemann Sums and Double integrals

Subsection 12.1.5 Summary

• Let \(f\) be a continuous function on a rectangle \(R = \{(x,y) : a \leq x \leq b, c \leq y \leq d\}\text{.}\) The double Riemann sum for \(f\) over \(R\) is created as follows.

  1. Partition the interval \([a, b]\) into \(m\) subintervals of equal length \(\Delta x = \frac{b-a}{m}\text{.}\) Let \(x_0\text{,}\) \(x_1\text{,}\) \(\ldots\text{,}\) \(x_m\) be the endpoints of these subintervals, where \(a = x_0\lt x_1\lt x_2 \lt \cdots \lt x_m = b\text{.}\)

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  2. Partition the interval \([c, d]\) into \(n\) subintervals of equal length \(\Delta y = \frac{d-c}{n}\text{.}\) Let \(y_0\text{,}\) \(y_1\text{,}\) \(\ldots\text{,}\) \(y_n\) be the endpoints of these subintervals, where \(c = y_0\lt y_1\lt y_2 \lt \cdots \lt y_n = d\text{.}\)

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  3. These two partitions create a partition of the rectangle \(R\) into \(mn\) subrectangles \(R_{ij}\) with opposite vertices \((x_{i-1},y_{j-1})\) and \((x_i, y_j)\) for \(i\) between \(1\) and \(m\) and \(j\) between \(1\) and \(n\text{.}\) These rectangles all have equal area \(\Delta A = \Delta x \cdot \Delta y\text{.}\)

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  4. Choose a point \((x_{ij}^*, y_{ij}^*)\) in each rectangle \(R_{ij}\text{.}\)

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  Then a double Riemann sum for \(f\) over \(R\) is given by

  \begin{equation*} \sum_{j=1}^n \sum_{i=1}^m f(x_{ij}^*, y_{ij}^*) \cdot \Delta A. \end{equation*}

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• With terms defined as in the Double Riemann Sum, the double integral of \(f\) over \(R\) is

  \begin{equation*} \iint_R f(x,y) \, dA = \lim_{m,n \to \infty} \sum_{j=1}^n \sum_{i=1}^m f(x_{ij}^*, y_{ij}^*) \cdot \Delta A. \end{equation*}

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• Interpretations of the double integral \(\iint_R f(x,y) \, dA\) are:

  • The volume of the solids the graph of \(f\) bounds above the \(xy\)-plane over the rectangle \(R\) minus the volume of the solids the graph of \(f\) bounds below the \(xy\)-plane under the rectangle \(R\text{;}\)

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  • Dividing the double integral of \(f\) over \(R\) by the area of \(R\) gives us the average value of the function \(f\) on \(R\text{.}\) If \(f(x, y) \geq 0\) on \(R\text{,}\) we can interpret this average value of \(f\) on \(R\) as the height of the box with base \(R\) that has the same volume as the volume of the surface defined by \(f\) over \(R\text{.}\)

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  • If \(f(x,y)\) is the density of a substance over some area \(R\text{,}\) then the double integral of \(f\) over \(R\) will measure the total amount of the substance in \(R\text{.}\)

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