Oil well burning in a 100 miles of the well( i think in a circle)

Question

When an oil well burns, sediment is carried up into the air by the flames and is eventually deposited on the ground. Less sediment is deposited further away from the well. Experimental evidence indicates that the density (in tons per square mile) at a distance r in any direction from the burning oil well is given by the formula S(r) = (7/(1+r^2)). In order to estimate how large a clean up crew to hire, a company must estimate how many tons of sediment must be removed within 100 miles of the well. Find and evaluate the integral that represents this total deposit.

Answer 1

I think you need an iterated integral to solve this problem, using polar co-ordinates.

The integrand is 7r dr dθ/(1+r²).

The limits of integration on the outer (θ) integral are 0 to 2π.

The limits on the inner (r) integral are 0 to 100

Bring the 7 outside and multiply numerator and denominator by 2.

Then use a u substitution u=1+r².

This will give you 7π ln(10001).

Answer 2

The problem is too long for a 5 minute video.

https://youtu.be/UBvL_pJQLTM