consider the deposit as a series of concentric rings of radius r and width dr. The amount of sediment in each ring is just the area times the density: 2πr dr * D(r)
So, add up all the rings to get the total mass:
m(r) = ∫[0,100] 2πr * 7/(1+r^2) dr
= 7π ∫ 2r/(1+r^2) dr
= 7π ln(1+r^2) [0,100]
= 7π ln(10001)
= 202.5 tons
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 give by the formula D(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. Round to nearest tenth.
2 answers
area of ring is pi*r^2, not 2pi*r (circumference).
1 answer