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- Consider the region S in the plane of all points which are be- tween the graphs of the parabola y = 9 − x^2 and the straight line y = x + 3. The region S is shown in the figure to the right.
1A. Set up the integral which represents the area of S if the corre- sponding approximations are rectangles each of whose base width is 4 x and each rectangle goes from a top curve down to a bottom curve. That is, the integral is of the form
dx.
1B. Evaluate the integral in (1A) to find that area. Show all your work.
2A. Set up the integral which gives the arc length of the curved part of the boundary of S, that is, the arc length of the graph of y = 9 − x^2 from x = −3 to x = 2.
2B. The integral in 2A is “doable” — the back of your book has a formula for the antiderivative you’d need. But it’s so complicated, and in practice, a good numerical approximation will do. Indeed, find the MID(50) approximation for the integral in 2A.
- Again, consider the region S from problem 1, of all points which are between the graphs of the parabola y = 9 − x^2 and the straight line y = x + 3. The region S is shown in the figure to the right.
3A. Set up the integral(s) which represents the volume of the solid of revolution obtained by revolving S around the line y = −1. (Do not evaluate the integral(s)).
3B. Now set up the integral(s) which represents the volume of the solid of revolution obtained by revolving S around the line x = −3. (Do not evaluate the integral(s)).
- The following table of velocities v(t) in feet per second at various times t for some moving object was recorded during an experiment: t 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5. v(t) -27 -21.1 -15.4 -10.3 -5.8 -2.0 1.3 4.0 6.4 8.4 10.
5A. What is the physical meaning of
1 v(t)^ dt?
5B. For the integral
1 v(t)^ dt^ in (5A), estimate LHS(n), RHS(n), TRAP(n) and MID(n) for the maximum possible number of subintervals n in each case, using only the information available in the table. Clearly label all your answers!
5C. The table suggests that v(t) is an increasing, concave-down function on [0, 5]. Suppose it is, and there was enough
information to find I =
0 v(t)^ dt^ for each of LHS(n), RHS(n), TRAP(n) and MID(n) with^ n^ = 25.^ From smallest to largest, put these numbers in order: I, LHS(n), RHS(n), TRAP(n) and MID(n).
6A. Find
1
cos (
x ) √ x
dx by the method of substitution, using appropriate notation throughout.
6B. In particular, what are the limits on the integral in (6A) after the appropriate substitution is made?
