Chapter 15, Problem 1CRE

To determine

Calculate the Riemann sum $S_{2,3}$ for the given integral using two choices of sample points:

a) Lower-left vertex

b) Midpoint of rectangle

Then calculate the exact value of the double integral.

Answer to Problem 1CRE

Solution:

(a) The Riemann sum $S_{2,3}$ for the given double integral using lower-left vertices is 240.

(b)The Riemann sum $S_{2,3}$ for the given double integral using midpoints is 510.

And the exact value of the double integral is 520.

Explanation of Solution

Given:

The integral: ${\displaystyle {\int }_1^4{\displaystyle {\int }_2^6{x}^{2}ydxdy}}$

Formulas:

$S_{m,n}={\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^{n}f(x_i,y_j)\Delta A}}$

Where

$\begin{array}{l}\Delta A=\Delta x\cdot \Delta y\\ \Delta x=\frac{b-a}{m}\text{and}\Delta y=\frac{d-c}{n}\end{array}$

Calculations:

From the given integral, we can observe that $2\le x\le 6$and $1\le y\le 4$. Since our aim is to find $S_{2,3}$, we need to divide the rectangle $[2,6]\times [1,3]$ into $2\times 3$ subrectangles. The length and width of each subrectangle are calculated as follows:

$\Delta x=\frac{6-2}{2}=\frac{4}{2}=2$

$\Delta y=\frac{4-1}{3}=\frac{3}{3}=1$

Therefore, the area of each subrectangle is $\Delta A=\Delta x\cdot \Delta y=2\cdot 1=2$.

The subrectangles are shown in Image 1.

Image 1:

(a) Using Lower-left vertex

Here, we use the lower-left vertices of each subrectangleto find the Riemann sum $S_{2,3}$. Notice that the lower-left vertices are $(2,1),(2,2),(2,3),(4,1),(4,2),\text{and }(4,3)$and are shown in Image 2.

Image 2:

Thus,

$\begin{array}{l}{\displaystyle {\int }_1^4{\displaystyle {\int }_2^6{x}^{2}ydx\text{}dy}}\\ ={\displaystyle \sum _{i=1}^2{\displaystyle \sum _{j=1}^{3}f(x_i,y_j)\Delta A}}\\ =[f(x_1,y_1)+f(x_1,y_2)+f(x_1,y_3)+f(x_2,y_1)+f(x_2,y_2)+f(x_2,y_3)]\Delta A\\ =[f(2,1)+f(2,2)+f(2,3)+f(4,1)+f(4,2)+f(4,3)]\cdot 2\\ =[2^2\cdot 1+2^2\cdot 2+2^2\cdot 3+4^2\cdot 1+4^2\cdot 2+4^2\cdot 3]\cdot 2\\ =[4+8+12+16+32+48]\cdot 2\\ =120\cdot 2\\ =240\end{array}$

(b) Using Midpoint of Rectangle:

Here, we use the midpoints of each subrectangle to find the Riemann sum $S_{2,3}$. Notice that the midpoints are $(3,1.5),(3,2.5),(3,3.5),(5,1.5),(5,2.5),\text{and (5,3}\text{.5)}$and are shown in Image 3.

Image 3:

Thus,

$\begin{array}{l}{\displaystyle {\int }_1^4{\displaystyle {\int }_2^6{x}^{2}ydx\text{}dy}}\\ ={\displaystyle \sum _{i=1}^2{\displaystyle \sum _{j=1}^{3}f\left(\overline{x_i},\overline{y_j}\right)\Delta A}}\\ =\left[f\left(\overline{x_1},\overline{y_1}\right)+f\left(\overline{x_1},\overline{y_2}\right)+f\left(\overline{x_1},\overline{y_3}\right)+f\left(\overline{x_2},\overline{y_1}\right)+f\left(\overline{x_2},\overline{y_2}\right)+f\left(\overline{x_2},\overline{y_3}\right)\right]\Delta A\\ =[f(3,1.5)+f(3,2.5)+f(3,3.5)+f(5,1.5)+f(5,2.5)+f(5,3.5)]\cdot 2\\ =[3^2\cdot 1.5+3^2\cdot 2.5+3^2\cdot 3.5+5^2\cdot 1.5+5^2\cdot 2.5+5^2\cdot 3.5]\cdot 2\\ =[13.5+22.5+31.5+37.5+62.5+87.5]\cdot 2\\ =255\cdot 2\\ =510\end{array}$

To calculate the exact value of the integral:

$\begin{array}{l}{\displaystyle {\int }_1^4{\displaystyle {\int }_2^6{x}^{2}ydx\text{}dy}}\\ ={\displaystyle {\int }_1^{4}ydy{\displaystyle {\int }_2^6{x}^{2}dx\text{}}}\\ ={\frac{y^2}{2}]}_1^4\cdot {\frac{x^3}{3}]}_2^6\\ =\left[\frac{4^2}{2}-\frac{1^2}{2}\right]\cdot \left[\frac{6^3}{3}-\frac{2^3}{3}\right]\\ =\left[8-\frac{1}{2}\right]\cdot \left[72-\frac{8}{3}\right]\\ =\frac{15}{2}\cdot \frac{208}{3}\\ =520\end{array}$

Conclusion:

Thus,

(a) The Riemann sum $S_{2,3}$ for the given double integral using lower-left vertices is 240.

(b)The Riemann sum $S_{2,3}$ for the given double integral using midpoints is 510.

And the exact value of the double integral is 520.