Sun K.

asked • 05/04/13# Find the mass of the plate with density?

Find the mass of the plate with density p(x, y)=ky, where k is a positive constant and R is the region bounded by y=x^2 and x=y^2.

## 2 Answers By Expert Tutors

Grigori S. answered • 05/04/13

Certified Physics and Math Teacher G.S.

The region R is bounded by two curves y=x^2 and y = sqrt(x). They intersect at two points x=0, y=0 , and x=1, y = 1. Mass is equal to the integral of p(x,y) dxdy = int kydxdy. limits of integration by x are variable:

x = y^2 for lower limit and x = sqrt(y) for upper limit (since 0<y<1, y^2 < sqrt(y)).. Integration by x gives us

mass = int ky(sqrt(y) - y^2)dy = 3k/20 = 0.15k.

The function p(x,y) represents the surface density, and these calculations are true for infinitely thin plate only.

Robert J. answered • 05/04/13

Certified High School AP Calculus and Physics Teacher

Since the density is the function of y, it is better to do integration over dy.

mass = density x area

So, the mass of the plate = ∫[0, 1] ky (√y - y^{2})dy = 3k/20 <==Answer

Robert J.

For your additional question:

The intersection of y=x^2 and x=y^2 is at x = 0, and 1. So, the lower limit is 0, and the upper limit is 1.

05/06/13

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Sun K.

Where did you get 0 to 1?

05/04/13