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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Electric charge is distributed over the rectangle 0 ≤ x ≤ 5, 2 ≤ y ≤ 5 so that the charge density at (x, y) is σ(x, y) = 2x + 4y (measured in coulombs per square meter). Find the total charge on the rectangle.

To determine

To find: The total charge over the given rectangular region.

Explanation

Given:

The rectangle R is 0x5,2y5.

The charge density at (x,y) is σ(x,y)=2x+4y.

Formula used:

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

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

Calculation:

Total charge Q on the given rectangle R is,

Q=Rσ(x,y)dA=0525(2x+4y)dydx

Integrate with respect to y,

Q=05[2xy+4y22]25dx=05[2xy+2y2]25dx=205[xy+y2]25dx

Apply the limit value of y,

Q=2

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