Chapter 14, Problem 1RE

The double integral over a region R in the $xy\text{-plane}$ is defined as

${\displaystyle \underset{R}{\iint }f\left(x,y\right)dA=\underset{n\to +\infty }{\lim }}{\displaystyle \underset{k=1}{\overset{n}{\sum }}f\left(x_k^{*},y_k^{*}\right)\nabla A_k}$

Describe the procedure on which this definition is based.

To determine

The procedure on which the double integral over a region $R$ in the $xy-\text{plane}$ is defined as ${\displaystyle \underset{R}{\iint }f\left(x,y\right)dA=\underset{n\to +\infty }{\lim }}{\displaystyle \sum _{k=1}^{n}f\left(x_k^{\ast },y_k^{\ast }\right)}\Delta A_k$ is based.

Answer to Problem 1RE

If a region divided into $n$ regions with lesser size and with approximately equal areas and the number of partitions increases, the approximation is usually better.

Explanation of Solution

Consider the given double integral over a region $R$ in the $xy-\text{plane}$ is defined as ${\displaystyle \underset{R}{\iint }f\left(x,y\right)dA=\underset{n\to +\infty }{\lim }}{\displaystyle \sum _{k=1}^{n}f\left(x_k^{\ast },y_k^{\ast }\right)}\Delta A_k$ .

First, compute a double integral ${\displaystyle \underset{R}{\iint }f\left(x,y\right)dA}$ over region $R$ .

Next, divide the region $R$ into $n$ regions with lesser size and with approximately equal areas $\Delta A_k$ .

Now, approximate the integral adding the products of the areas multiplied by the values of the given function $f\left(x,y\right)$ evaluated at some point within this region $\left(x_k^{\ast },y_k^{\ast }\right)$ .

${\displaystyle \underset{R}{\iint }f\left(x,y\right)dA\approx \underset{n\to +\infty }{\lim }}{\displaystyle \sum _{k=1}^{n}f\left(x_k^{\ast },y_k^{\ast }\right)}\Delta A_k$

When the number of partitions increases, the approximation is usually better.

Therefore, if the sum converges, when the number of partitions tends to infinity, then the exact result is obtained.

${\displaystyle \underset{R}{\iint }f\left(x,y\right)dA=\underset{n\to +\infty }{\lim }}{\displaystyle \sum _{k=1}^{n}f\left(x_k^{\ast },y_k^{\ast }\right)}\Delta A_k$