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4. A thin wire has the shape of the first-quadrant part of the circle with center the origin andra 5. If the density function is 8(x, y) = 2xy , find the mass of the wire.

User Choxsword
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1 Answer

3 votes

Answer:

the mass of the wire is 125/4.

Explanation:

To find the mass of the wire, we need to integrate the density function over the wire. Since the wire has the shape of the first-quadrant part of the circle with center at the origin and radius 5, we can write its equation as:

x^2 + y^2 = 25

Solving for y, we get:

y = sqrt(25 - x^2)

Since the wire is thin, we can assume that its thickness is negligible, so we can treat it as a 2D object. The mass of an infinitesimal element of the wire can be written as:

dm = density * dA

where dA is the infinitesimal area of the element. In polar coordinates, we have:

x = r cos(theta)

y = r sin(theta)

dA = r dr dtheta

Substituting and simplifying, we get:

dm = 2r^3 sin(theta) cos(theta) dr dtheta

To find the total mass of the wire, we need to integrate dm over the first-quadrant part of the circle:

m = ∫∫ 2xy dA

where the limits of integration are:

0 ≤ r ≤ 5

0 ≤ theta ≤ π/2

Substituting the expressions for x and y, we get:

m = ∫[0,π/2] ∫[0,5] 2r^3 sin(theta) cos(theta) dr dtheta

Integrating with respect to r first, we get:

m = ∫[0,π/2] sin(theta) cos(theta) ∫[0,5] 2r^3 dr dtheta

m = ∫[0,π/2] sin(theta) cos(theta) [r^4]_0^5 dtheta

m = ∫[0,π/2] 125 sin(theta) cos(theta) dtheta

m = 125/2 [sin^2(theta)]_0^π/2

m = 125/4

Therefore, the mass of the wire is 125/4.

User Chris Simpkins
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