4.2 Double Riemann Sums and Double Integrals over Rectangles

Double Riemann Sums and Double Integrals over Rectangles

Introduction

Double Riemann Sums over Rectangles

Double Riemann Sums and Double Integrals

Interpretation of Double Riemann Sums and Double integrals

A plot of $f$ for inputs in the interval $[0,2]$ is shown in . 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 $[0,2]$. You can use whichever point you want on each subinterval to evaluate the height the of the rectangles.

To understand the numerical calculations involved in the classic calculus approach for a double integral, it is most important to understand the region of integration. Thus, we will not look at a graph of $z=f(x,y)$. Instead, we will stay focused in the $xy$-plane. On the axes below, outline the rectangular region $R$ that corresponds to the region of integration.

Because all of the regions and subregions we are considering in this section are rectangles, we can break up the $x$ and $y$ coordinates into pieces separately. For this activity, break the interval of $x$-coordinates into four equally-sized subintervals and break the interval of $y$-coordinates into three equally-sized subintervals.

What is the length of each subinterval in the $x$-direction? How about in the $y$-direction?

Let $S_i$ be the $i$-th subinterval for $x$. We want to state the endpoints of each of the $S_i$. The first subinterval, $S_1$, will go from $x_0$ to $x_1$, the second subinterval, $S_2$, will go from $x_1$ to $x_2$, the third subinterval, $S_3$, will go from $x_2$ to $x_3$, and the fourth subinterval, $S_4$, will go from $x_3$ to $x_4$.

Give the values for $x_0$, $x_1$, $x_2$, $x_3$, and $x_4$ and add these as tick marks on the $x$-axis of the graph in to make sure your subintervals are equally sized.

Let $T_i$ be the $i$-th subinterval for $y$. Give the values for $y_0$, $y_1$, $y_2$, and $y_3$ and add these as tick marks on the $y$-axis of the graph in to make sure your subintervals are equally sized.

We will use the subintervals in the $x$- and $y$-coordinates to specify smaller rectangles into which $R$ is divided to compute an approximation. Let $R_{i j}$ be the rectangle corresponding to $S_i \times T_j$. - How many smaller rectangles are there in this partition? - Outline each of the smaller rectangles on the graph in and label each rectangle as either $R_{1 1}, R_{1 2} , ...$. - Since each smaller rectangle $R_{i j}$ has the same area, let $\Delta A$ denote the area of each of these smaller rectangles. What is $\Delta A$?

To find the volume of the rectangular prisms over each $R_{i j}$, we must pick a point in each subrectangle at which to evaluate $f$, which will be the height of the rectangular prism. For this activity, we will use the upper-right corner of each subrectangle as the designated point.

State the point at the upper right of each smaller rectangle and evaluate $f$ at each of these points.

Write a sentence about why the volume of each rectangular prism used for this approximation is

. Remember that $x_i$ and $y_j$ are the points from parts c and d of this activity. Write a couple sentences about how you would find an approximation of the volume under the surface $z=f(x,y)$ over the region $R$. (Do not do calculation, but rather explain what calculation is being done.)

We chose the upper-right point of each subrectangle $R_{i j}$ to find the height of each rectangular prism used in the approximation. 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 $R_{i j}$. 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.

Sketch the region $R$ in the plane partitioned as described above.

Calculate the double Riemann sum using the given partition of $R$ and the values of $f$ in the upper right corner of each subrectangle.

Use geometry to calculate the exact value of $\displaystyle\iint_R f(x,y) \, dA$ and compare it to your approximation. Write a sentence to describe one way to obtain a better approximation using the given data.

Draw a picture of $R$. Partition $[0,2]$ into 2 subintervals of equal length and the interval $[1,3]$ into two subintervals of equal length. Draw these partitions on your picture of $R$ and label the resulting subrectangles using the labeling scheme we established in the definition of a double Riemann sum.

For each $i$ and $j$, let $(x_{ij}^*, y_{ij}^*)$ be the midpoint of the rectangle $R_{ij}$. Identify the coordinates of each $(x_{ij}^*, y_{ij}^*)$. Draw these points on your picture of $R$.

Calculate the Riemann sum

using the partitions we have described. If we let $(x_{ij}^*, y_{ij}^*)$ be the midpoint of the rectangle $R_{ij}$ for each $i$ and $j$, then the resulting Riemann sum is called a midpoint sum.

Explain the meaning of the sum you just calculated in terms of each of the interpretations in Proposition 4.2.1