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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Electric charge is distributed over the disk x2 + y2 ≤ 1 so that the charge density at (x, y) is σ ( x , y ) = x 2 + y 2 (measured in coulombs per square meter). Find the total charge on the disk.

To determine

To find: The total charge on the given disk.

Explanation

Given:

The region D is the disk x2+y21.

The charge density at (x,y) is σ(x,y)=x2+y2.

Formula used:

The total charge Q on the given rectangle R is, Q=Rσ(x,y)dA.

Here, the charge density is given by σ(x,y).

If f is a polar rectangle R given by 0arb,αθβ, where 0βα2π, then,

Rf(x,y)dA=αβabf(rcosθ,rsinθ)rdrdθ (1)

If g(x) is the function of x and h(y) is the function of y then,

abcdg(x)h(y)dydx=abg(x)dxcdh(y)dy (2)

Calculation:

Total charge Q on the given rectangle R is,

Q=Rσ(x,y)dA=0102πx2+y2dydx

Substitute x=rcosθ,y=sinθ and convert the coordinates into polar coordinates

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Chapter 15 Solutions

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