## Calculus (3rd Edition)

Published by W. H. Freeman

# Chapter 16 - Multiple Integration - 16.1 Integration in Two Variables - Exercises - Page 846: 3

#### Answer

Choice 1. Use midpoints as sample points (Figure 16 (A)) ${S_{3,2}} = 42$ Choice 2. Use the sample points as in Figure 16 (B) ${S_{3,2}} = 43.5$

#### Work Step by Step

Compute the Riemann sums for the double integral $\mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal R}^{} f\left( {x,y} \right){\rm{d}}A$, where ${\cal R} = \left[ {1,4} \right] \times \left[ {1,3} \right]$. From Figure 16, we obtain the grid: $N \times M = 3 \times 2$. Using the regular partition, we get the dimensions of the subrectangles: $\Delta x = \frac{{4 - 1}}{3} = 1$, ${\ \ \ \ }$ $\Delta y = \frac{{3 - 1}}{2} = 1$ The Riemann sum ${S_{3,2}}$ is given by ${S_{3,2}} = \mathop \sum \limits_{i = 1}^3 \mathop \sum \limits_{j = 1}^2 f\left( {{P_{ij}}} \right)\Delta {x_i}\Delta {y_j} = \mathop \sum \limits_{i = 1}^3 \mathop \sum \limits_{j = 1}^2 f\left( {{P_{ij}}} \right)$ Choice 1. Use midpoints as sample points (Figure 16 (A)) ${S_{3,2}} = f\left( {\frac{3}{2},\frac{3}{2}} \right) + f\left( {\frac{5}{2},\frac{3}{2}} \right) + f\left( {\frac{7}{2},\frac{3}{2}} \right) + f\left( {\frac{3}{2},\frac{5}{2}} \right) + f\left( {\frac{5}{2},\frac{5}{2}} \right) + f\left( {\frac{7}{2},\frac{5}{2}} \right)$ $= \frac{9}{2} + \frac{{13}}{2} + \frac{{17}}{2} + \frac{{11}}{2} + \frac{{15}}{2} + \frac{{19}}{2}$ ${S_{3,2}} = 42$ Choice 2. Use the sample points as in Figure 16 (B) The sample points are: $\left( {\frac{3}{2},\frac{3}{2}} \right)$, $\left( {2,1} \right)$, $\left( {\frac{7}{2},\frac{3}{2}} \right)$, $\left( {2,3} \right)$, $\left( {\frac{5}{2},\frac{5}{2}} \right)$, $\left( {4,3} \right)$. So, ${S_{3,2}} = f\left( {\frac{3}{2},\frac{3}{2}} \right) + f\left( {2,1} \right) + f\left( {\frac{7}{2},\frac{3}{2}} \right) + f\left( {2,3} \right) + f\left( {\frac{5}{2},\frac{5}{2}} \right) + f\left( {4,3} \right)$ $= \frac{9}{2} + 5 + \frac{{17}}{2} + 7 + \frac{{15}}{2} + 11$ ${S_{3,2}} = 43.5$

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