Prof. Brian Harbourne's old exam

• [1] Use the substitution u = 4 - 2x to convert the integral, from x = 0 to x = 1 of 2x(4 - 2x)^1/2 with respect to x, into an integral in terms of u only. You do not need to evaluate the integral.
• [2] Let R be the region bounded by the curves y = x^1/2, y = 2 - x, and y = 0. Sketch the region R, and express its area in terms of one or more integrals. You do not need to evaluate the integrals.
• [3] Let R be the region bounded by the curves y = 1, x = 1, and y = 100/x². Sketch the region R, and write down an integral for the volume of the solid formed by revolving R about the y-axis. You donot need to evaluate the integral.
• [4] A force of 10 pounds stretches a spring 2 inches. Set upan integral for the work done in stretching the spring 3 inches beyond its natural length. You do not need to evaluate the integral, but indicate what units of work the integral will evaluate to.
• [5] Baby bear's porridge is 150F when Mother bear sets it out at 9:00 am. The den is a comfy (for bears) 50F. At 9:10 am, the porridge is 120F, still too hot for Baby bear, who likes porridge to be 99F. What time will it be when Baby bear's porridge is just right? (Assume the porridge cools according to Newton's Law of cooling.)