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Chapter 5: Q34P (page 248)

A dielectric lamina with charge density proportional to y covers the area between the parabola y=16路x2 and the x axis. Find the total charge.

Short Answer

Expert verified

The total charge of the dielectric lamina is,546.13K.

Step by step solution

01

Definition of double integral and mass formula

The double integral of f(x,y)over the area Ain the (x,y) plane as the limit of this sum, and we write it as

Using the double integral the mass of the lamina:localid="1658895349906" M=鈭�鈭�A蚁诲础.

02

Diagram of the dielectric lamina

The figure of the dielectric lamina with charge density proportional to y covers the area between the parabola y=16-x2 and x theaxis.

03

Calculation of the charge of the dielectric lamina

The total charge of the lamina is given by,

Q=鈭�鈭�A蚁诲础=鈭�-44鈭�016-x2Kydydx=K鈭�-4412y2016-x2dx=K2鈭�-4416-x22dx=K2鈭�-44256-32x2+x4dx=K2256x-323x3+15x5-44=546.13K

where K is the proportionality constant.

Therefore, the total charge is,546.13K.

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Most popular questions from this chapter

Find the area of the plane x-2y+5z=13 cut out by the cylinder x2+y2=9.

In Problems 17 to 30, for the curve y=x, betweenx=0andx=2, find:

The moments of inertia about the x axis of a lamina in the shape of the plane area under the curve;

鈭�x=01鈭�y=03-3xdydx

(a) Write a triple integral in cylindrical coordinates for the volume of the part of a ball between two parallel planes which intersect the ball.

(b) Evaluate the integral in (a). Warning hint: Do the r and胃integrals first.

(c) Find the centroid of this volume.

In the integral

I=鈭�0鈭�鈭�0鈭�x2+y2(x2+y2)2e-2xydxdy.

Make the change of variables

u=x2-y2v=2xy

And evaluate I. Hint: Use (4.8) and the accompanying discussion.
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