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Density and mass Suppose a thin rectangular plate,
Chapter 11, Problem 47E(choose chapter or problem)
Density and mass Suppose a thin rectangular plate, represented by a region R in the xy-plane, has a density given by the function \(\rho(x, y)\); this function gives the area density in units such as \(\mathrm{g} / \mathrm{cm}^{2}\).The mass of the plate is \(\iint_{R} \rho(x, y) \ d A\). Assume that \(R=\{(x, y): 0 \leq x \leq \pi / 2,0 \leq y \leq \pi\}\) and find the mass of the plates with the following density functions.
a. \(\rho(x, y)=1+\sin x\)
b. \(\rho(x, y)=1+\sin y\)
c. \(\rho(x, y)=1+\sin x \sin y\)
Questions & Answers
QUESTION:
Density and mass Suppose a thin rectangular plate, represented by a region R in the xy-plane, has a density given by the function \(\rho(x, y)\); this function gives the area density in units such as \(\mathrm{g} / \mathrm{cm}^{2}\).The mass of the plate is \(\iint_{R} \rho(x, y) \ d A\). Assume that \(R=\{(x, y): 0 \leq x \leq \pi / 2,0 \leq y \leq \pi\}\) and find the mass of the plates with the following density functions.
a. \(\rho(x, y)=1+\sin x\)
b. \(\rho(x, y)=1+\sin y\)
c. \(\rho(x, y)=1+\sin x \sin y\)
ANSWER:Solution 47EStep 1:In this problem we have to find the mass of the plates with the given density functions.Given: The mass of the plate is Assume that R = {(x, y): 0 x /2, 0 y }a. (x, y) = 1+sin xThus mass of the plate with density function (x, y) = 1+sin x is given by … (1)First evaluate the inner integral (Since and )Thus (1) becomes, Thus mass of the plate with density function (x, y) = 1+sin x is .