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Math 152 Math 152 - Final Exam Practice #1 1. Find the volume of the solid formed by rotating the region below about the line y = 1. 1 Math 152 2 2. The Sudbury Neutrino Observatory is a large spherical tank with a 6 meter radius filled with heavy water. How much work does is take to pump all the water out of the top of the tank? (Heavy water has a density of 1107 kilograms per cubic meter.) Math 152 3 3. The curve pictured below is the curve x = 3t − t3 , y = 3t2 √ √ for t-values from − 3 to 3. That is the total arclength of the curve? Math 152 4 4. Recall that in a circuit containing a battery with voltage E, and resistor with resistance R and a capacitor with capacitance C, the charge Q on the capacitor is a function of time t that satisfies the equation dQ R + CQ = E. dt As the circuit operates, the battery warms up. This causes the voltage to vary with time. If R = 2 ohms, C = 3 f arads, E(t) = 12 t + 4 volts, find the charge Q on the capacitor at time t if the capacitor starts with no charge, i.e. Q(0) = 0. Math 152 5 5. Solve the differential equation dy xy = . dx (x − 2)(x + 4) Math 152 6 6. Consider the sprial r = ln θ. (a) Write an integral to express the arclength of the curve from θ = 1 to θ = 3. (Do NOT try to evaluate the integral!) (b) Approximate the arclength from part (a) using the trapezoid method and ∆θ = 21 . Math 152 7 7. Calculate Z 3 4.5 p 6x − x2 dx.