Çözüldü Integral Application to Physics

Taking Earth's radius as r from its center and a very small thickness of it as dr, the differential volume is dV = 4π(r^2)dr, and the density as a function of radius is ρ(r). Then the differential mass is dm = [ ρ(r) ]·dV = [ ρ(r) ]·4π(r^2)dr .
So, the actual mass is found by integrating dm from its center to surface by using MKS units;
m = (lower limit r1 = 0, upper limit r2 = 6.37·10^(6), ∫ [ ρ(r) ]·4π(r^2) dr
m = (lower limit r1 = 0, upper limit r2 = 6.37·10^(6), 4π· ∫ [ ρ(r) ]·(r^2) dr....(I)
Now that the density linearly decreases, then its line equation is supposed to have a negative slope. Therefore, the equation of the density function is, with positive real coefficients m and r;
ρ(r) = -m·r + n....(II)
Recalling from analytic geometry, the equation of the line passing through two points;
[ ρ(r) - ρ(0) ] / [ ρ(0) - ρ(6370 km) ] = (r - 0) / (0 - 6370 km)
[ ρ(r) - 12500 km ] / [ 12500 km - 950 (kg / m^3) ] = (r - 0) / (0 - 6370 km) and organizing in the form of ρ(r) = -m·r + n, converting from km to meter unit and writing in the scientific exponential form;
ρ(r) = -{ 11550 / [ 6.37·10^(6) ] }·r + 12500
ρ(r) ≈ -1.81·[ 10^(-3) ]·r + 12500....(III)
Comparing (III) to (II), the m and n coefficients are m = 1.81·[ 10^(-3) ]....(IV), n = 12500....(V)
Completing the integration at (I) using (II);
m ≈ (lower limit r1 = 0, upper limit r2 = 6.37·10^(6), 4π·∫ (-m·r + n)(r^2) dr
m ≈ (lower limit r1 = 0, upper limit r2 = 6.37·10^(6), 4π·∫ [ -m·(r^3) + n·(r^2) ] dr
m ≈ (lower limit r1 = 0, upper limit r2 = 6.37·10^(6), 4π·|-m·(r^4) / 4 + n·(r^3) / 3|
m ≈ 4π·{ -m·[ 6.37·10^(6) ]^4 / 4 + n·[ 6.37·10^(6) ]^3 / 4 }....(VI)
Plugging the values of m and n at (IV) and (V) respectively;
m ≈ 4π·{ -1.81·[ 10^(-3) ]·(6.37·10^6)^4 / 4 + 12500·(6.37·10^6)^3 / 3 }
m ≈ 4.17·[ 10^(24) ] kg

Note: Since the real mass of the Earth is 5.97·[ 10^(24) ] kg, it would be correct to conclude that the real change of the density is different than the linear decrease assumption supposed in this homework.
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Sorunun Yedeği.: https://i.ibb.co/r5wX8dq/HW-3.jpg