Today, we will look at several additional applied problems where the quantity to be computed can be approximated by Riemann sums, and computed by an integral. For each problem, carry out the subdivide, approximate, sum procedure to set up an integral computing the desired quantity.
A) A large bowl has the shape of the solid obtained by rotating the region bounded by y = x4+1 and y = 0, between x=0 and x=1 (x in feet), about the y-axis. Find the volume of the bowl, and the volume of liquid the bowl can hold. (Hint: How could you "slice" to get manageable cross-sections? What are the cross-sections? What is their area?)
Answer: volume of the bowl itself: 4 Pi/3 ft3, volume the bowl holds: 2 Pi /3 ft3.
B) Your car, moving at a speed of v miles per hour, achieves 25 + 0.1v miles per gallon of regular gas, for speeds between v = 20 and v = 60 miles per hour. On a trip, your speed as a function of time is v = 50 t/(1 + t). How many gallons of gas did you use between t = 2 and t = 3 hours? (Hint: In a time interval Delta t hours long, you travel roughly v Delta t miles, and use gas accordingly. You can measure gas consumption in gallons per mile or miles per gallon. How many miles per gallon are you using if you use .25 gallons per mile, for example?)
Answer: About 1.25 gallons.
C) A bagel (mathematically, a torus) is obtained by rotating the region inside the circle x2 + (y-a)2 = b2 (a > b) about the x-axis. Find the volume of the bagel in terms of a and b.
Answer: (2 Pi a)(Pi b2) = 2 Pi2 a b2.
Group writeups due Friday, November 7.