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Chapter 13: Q. 24 (page 1004)

Evaluate the sums in Exercises 23鈥�28.
${鈭�}_{j = 1}^{3} {鈭�}_{k = 1}^{2} k^{j}$

Short Answer

Expert verified

The value of summation is $17$

Step by step solution

01

Step 1. Given information

An expression is given as ${鈭�}_{j = 1}^{3} {鈭�}_{k = 1}^{2} k^{j}$

02

Step 2. Evaluating summation

First separate out the terms then simplify,

${鈭�}_{j = 1}^{3} {鈭�}_{k = 1}^{2} k^{j} = {鈭�}_{j = 1}^{3} ({鈭�}_{k = 1}^{2} k^{j}) {鈭�}_{k = 1}^{2} k^{j} = 1^{j} + 2^{j} 鈬� (1^{1} + 2^{1}) + (1^{2} + 2^{2}) + (1^{3} + 2^{3}) = 1 + 2 + 1 + 4 + 1 + 8 = 17$

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Most popular questions from this chapter

Let $惟$ be a lamina in the xy-plane. Suppose $惟$ is composed of n non-overlapping lamin忙 role="math" localid="1650321722341" $惟_{1}, 惟_{2}, 鈥�, 惟_{n} .$ Show that if the masses of these lamin忙 are $m_{1}, m_{2}, 鈥�, m_{n}$ and the centers of masses are $({\overset{炉}{x}}_{1}, {\overset{炉}{y}}_{1}), ({\overset{炉}{x}}_{2}, {\overset{炉}{y}}_{2}), 鈥�, ({\overset{炉}{x}}_{n}, {\overset{炉}{y}}_{n}),$ then the center of mass of $惟$ is $(\overset{炉}{x}, \overset{炉}{y}),$ where

$\overset{炉}{x} = \frac{{鈭�}_{k = 1}^{n} m_{k} {\overset{炉}{x}}_{k}}{{鈭�}_{k = 1}^{n} m_{k}} and \overset{炉}{y} = \frac{{鈭�}_{k = 1}^{n} m_{k} {\overset{炉}{y}}_{k}}{{鈭�}_{k = 1}^{n} m_{k}} .$

In Exercises, let $S = \{(x, y) 鈭� x^{2} + y^{2} 鈮� 1 and x 鈮� 0\} .$

If the density at each point in S is proportional to the point鈥檚 distance from the origin, find the center of mass of S.

Let $惟$ be a lamina in the xy-plane. Suppose $惟$ is composed of two non-overlapping lamin $惟_{1}$ and $惟_{2}$ , as follows:

Show that if the masses and centers of masses of $惟_{1}$ and $惟_{2}$ are $m_{1}$ and $m_{2},$ and $({\overset{炉}{x}}_{1}, {\overset{炉}{y}}_{1}) and ({\overset{炉}{x}}_{2}, {\overset{炉}{y}}_{2})$ respectively, then the center of mass of $惟$ is $(\overset{炉}{x}, \overset{炉}{y}),$ where

$\overset{炉}{x} = \frac{m_{1} {\overset{炉}{x}}_{1} + m_{2} {\overset{炉}{x}}_{2}}{m_{1} + m_{2}} and \overset{炉}{y} = \frac{m_{1} {\overset{炉}{y}}_{1} + m_{2} {\overset{炉}{y}}_{2}}{m_{1} + m_{2}}$

Find the volume between the graph of the given function and the xy-plane over the specified rectangle in the xy-plane

$f (x, y) = \frac{y}{x} e^{y} W j e r e R = {(x, y) | 1 鈮� x 鈮� e, 0 鈮� y 鈮� 2}$

What is the difference between a double integral and an iterated integral?

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