   Chapter 14, Problem 36RE

Chapter
Section
Textbook Problem

Finding the Mass of a LaminaIn Exercises 35 and 36, find the mass of the lamina described by the inequalities, given that its density is p ( x , y ) = x + 3 y . x ≥ 0 ,     0 ≤ y ≤ 4 − x 2

To determine

To calculate: The mass of the lamina described by the inequalities x0,0y4x2, where Density is given by ρ(x,y)=x+3y.

Explanation

Given:

Density is ρ(x,y)=x+3y.

Formula used:

m=Aρ(x,y)dA

Graph:

Plot the graph for inequalities x0,0y4x2

First inequality x0 represents graph will be on right hand side of y axis.

From second inequality 0y4x2, highest limit of y is 4x2 andlower limit of y is zero

⇒, y=4x2

Squaring both the sides,

y2=4x2

Add x2 both the side:

y2+x2=4x2+x2y2+x2=4

Above equation is the equation of circle having radius r=2

Draw the respective graph:

Calculation:

From the graph,

m=x=0x=2y=0y=4x2(x+3y)dydx

Solve using Polar coordinates

m=x=0x=2y=0y=4x2(x+3y)dydx=θ=0θ=π2r=0r=2(rcosθ+3rsinθ)rdrdθ=

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