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# AP Calculus AB March 16, 2009

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Applications of integrals review.

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### AP Calculus AB March 16, 2009

1. 1. Gearing up for the test Gearing Up by icathing
2. 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. 3. (a) Using this data and a Riemann sum, estimate the total population living in the 8 mile radius. HOMEWORK
4. 4. (a) Using this data and a Riemann sum, estimate the total population living in the 8 mile radius. HOMEWORK
5. 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.
6. 6. THE REST OF THESE ARE HOMEWORK
7. 7. These are the correct answers, although they are not necessarily in the correct order. ;-)
8. 8. 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]
9. 9. 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.
10. 10. 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.
11. 11. 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]