Answer:
Therefore the mass of the of the oil is 409.59 kg.
Step-by-step explanation:
Let us consider a circular disk. The inner radius of the disk be r and the outer diameter of the disk be (r+Δr).
The area of the disk
=The area of the outer circle - The area of the inner circle
= [tex]\pi (r+\triangle r)^2- \pi r^2[/tex]
[tex]=\pi [r^2+2r\triangle r+(\triangle r)^2-r^2][/tex]
[tex]=\pi [2r\triangle r+(\triangle r)^2][/tex]
Since (Δr)² is very small, So it is ignorable.
∴[tex]A=2\pi r\triangle r[/tex]
The density [tex]\delta (r)= \frac{40}{1+r^2}[/tex]
We know,
Mass= Area× density
[tex]=(2r \pi \triangle r)(\frac{40}{1+r^2}})[/tex]
Total mass [tex]M=\sum_{i=1}^n \frac{80r_i\pi }{1+r^2}\triangle r_i[/tex]
Therefore
[tex]\sum_{i=1}^n \frac{80r_i\pi }{1+r^2}\triangle r_i=\int_0^5 \frac{80r\pi }{1+r^2}dr[/tex]
[tex]=40\pi[ln(1+r^2)]_0^5[/tex]
[tex]=40\pi [ln(1+5^2)-ln(1+0^2)][/tex]
[tex]=40\pi ln(26)[/tex]
= 409.59 kg (approx)
Therefore the mass of the of the oil is 409.59 kg.
