Applications of integrals review.

1. Gearing up
for the test
Gearing Up by icathing

2. Greater Boston can be approximated by a semicircle of radius 8 miles
with its centre on the coast. Moving away from the centre along a
radius, the population density is constant for the first mile. Beyond that,
the density starts to decrease according to the data given in the table,
where ρ(r), thousands/mile2 , is the population density at a distance r
miles from the centre.
(a) Using this data and a Riemann sum, estimate the total
population living in the 8 mile radius.
(b) Determine a possible formula for ρ(r). Use this formula to make
another estimate of the population.

3. (a) Using this data and a Riemann sum, estimate the total
population living in the 8 mile radius.
HOMEWORK

4. (a) Using this data and a Riemann sum, estimate the total
population living in the 8 mile radius.
HOMEWORK

5. Greater Boston can be approximated by a semicircle of radius 8 miles
with its centre on the coast. Moving away from the centre along a
radius, the population density is constant for the first mile. Beyond that,
the density starts to decrease according to the data given in the table,
where ρ(r), thousands/mile2 , is the population density at a distance r
miles from the centre.
(b) Determine a possible formula for ρ(r). Use this formula to make
another estimate of the population.

8. THE
REST
OF
THESE
ARE
HOMEWORK

13. These are the correct answers,
although they are not necessarily
in the correct order. ;-)

14. Now let's practice what we've learned ...
Find the average value of ƒ on the given interval.
Find c such that ƒ = ƒ(c).
ave
Sketch the graph of ƒ and a rectangle whose area is the same as
the area under the graph of ƒ.
ƒ(x) = 2x, [0, 3]

15. Now let's practice what we've learned ...
Find the average value of ƒ on the given interval.
Find c such that ƒ = ƒ(c).
ave
Sketch the graph of ƒ and a rectangle whose area is the same as
the area under the graph of ƒ.

16. Consider the region P bounded by the
graph of the function ƒ between x=-8
and x=-5.
Set up, but do not evaluate, the integral
that represents the volume of the solid
generated by revolving P about:
(a) the y-axis. (b) the line x=-10. (c) the line x=3.

17. Now let's practice what we've learned ...
Find the average value of ƒ on the given interval.
Find c such that ƒ = ƒ(c).
ave
Sketch the graph of ƒ and a rectangle whose area is the same as
the area under the graph of ƒ.
ƒ(x) = x 2 + 2x - 5, [-2, 2]