Question

Find the mass of a lamina in the form of an ellipse `x^2/a^2+y^2/b^2=1`, If the density at any point varies as the product of the distance from the
The axes of the ellipse.

Answer 1

Mass of lamina is given by , M = ∫∫𝒓 𝒅𝒙 𝒅𝒚
r is the density function r =k xy

Ellipse eqn is : `x^2/a^2+y^2/b^2=1`

`0<=y<=bsqrt(a^2-x^2)/a`

𝟎 ≤ 𝒙 ≤ 𝒂

`thereforeM=4int_0^aint_0^(bsqrt(a^2-x^2)/a)kxydydx`

`=4kint_0^ax[y^2/2]_0^(bsqrt(a^2-x^2)/a)dx`

`=2kint_0^axb^2/a^2(a^2-x^2)dx`

`=(2kb^2)/a^2int_0^a(a^2x-x^3)dx`

`=(2kb^2)/a^2[(a^2x^2)/2-x^4/4]_0^a`

`thereforeM=(ka^2b^2)/2`