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Question
Find the mass of lamina bounded by the curves ๐ = ๐๐ − ๐๐ and ๐ = ๐๐ if the density of the lamina at any point is given by `24/25 xy`
Solution
The curve `y=x^2-3x` i.e `y+9/4=(x-3/2)^2`parabola intersecting the x-axis in x = 0 and x = 3. The line y = 2x intersects this parabola at x2 – 3x = 2x i.e. x2 – 5x = 0 i.e. at x = 0, x = 5. Therefore, points of intersection are (0,0) and (5,10). The surface density is ρ = (24/25)xy.
Taking the elementary strip parallel to the y-axis, on the strip y varies from ๐ฆ = ๐ฅ2 − 3๐ฅ to ๐ฆ = 2๐ฅ and then x varies from x = 0 to x = 5.
∴ Mass of lamina = `int_0^5 int_(x^2-3x )^(2x) xydxdy`
=`24/25 int_0^5 x[y^2/2]_(x^2-3x)^(2x) 24/25 xydxdy`
`=24/50int_0^5 4x^3-x(x^4-6x^3+9x^2) dx`
=`24/50int_0^5-5x^3+6x^4-x^5 dx`
=`24/50[-x^6/b+ (6x^5)/5-(5x^4)/4]_0^5`
=`24/50. 5^4 [-25/6+6-5/4]`
=`24/50. 5^4. 7/12`
∴ Mass of lamina = 175.
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