S1R5.MWS Instructions: Only the right side pages will be graded. Do all scratch work on the left side and then neatly present your solution on the right. Your solution must include not only the answer but also all the work that is necessary to obtain the answer. 1. Find the area of the region bounded by the curves y=x and y=x^2. 2. Find the area of the region bounded by y=x/2 and y^2=8-x. 3. What is the average value of y=1/(1+x^2) for x in the interval [1/sqrt(3),sqrt(3)? 4. The graph of y=sqrt(sin(x)) for x in the interval [0,Pi is rotated about the x-axis. What is the volume of the solid of revolution? 5. The region in the first quadrant that lies under y=4sqrt(x) and above y=(x/2)^3 is rotated about the y-axis. What is the volume of the solid of revolution? 6. The shaded region in the accompanying figure has a boundary that consists of two line segments and an arc of the circle centered at (0,3). If this figure is rotated about the y-axis, then what is the volume of the resulting solid of revolution ? 7. The "triangular" region bounded by y=x^3, y=8, and x=0 is rotated about x=2. Express the volume of the solid of revolution by means of an integral. You do not need to evaluate the integral. 8. A spring is stretched xi meters beyond its equilibrium position. The number xi is not noted but it is noted that a 3N force is needed to maintain the spring at that position. It is also noted that 5J of work must be done to stretch the spring an additional 1m (that is, from the unrecorded stretched position). Find the value of the spring constant and determine the value of xi. 9. A 30ft cable is suspended from the top of a building. The cable is uniform and weighs 1lb per foot. A bucket of cement is attached to the end of the cable. The cable is cranked so that the bucket rises 20ft. If doing so requires 2000ft-lbs work, then how much does the bucket of cement weigh? 10. Evaluate int(x*sin(x),x=0..Pi. (You must show your work. A calculator-generated numerical answer is not acceptable.) 11. Evaluate int(ln(x^(3)),x=1..exp(1). (You must show your work. A calculator-generated numerical answer is not acceptable.) 12. Suppose that n is a constant different from -1. Evaluate int(x^n*ln(x),x.