ams per Liter. Given that the concentration of salt in the water varies linearly as a function ρ(y) of y, write down a formula for ρ(y). (d) Determine the amount of salt in the representative slice at height y and use your result to write down an integral the value of which is equal to the amount of salt in the solution. Finally evaluate the integral to get an approximate value for the amount of salt in the water and include units in your answers.
1. Let R be the region in the plane bounded by the y-axis, the curve y = ex , and the line y = 6. Suppose that the units of the x- and y-axes are measured in centimeters.
(a) Find and evaluate an expression for the volume of the solid obtained by revolving the region R around the y-axis. Show your work by sketching the solid, drawing and labelling a representative slice at height y of width ∆y and writing down a formula for the volume of the slice as a function of y and ∆y. Include units in your final answer.
(b) If the solid has a uniform density of 1 g/cm3 , find and evaluate an expression for the y-coordinate of the center of mass of the solid. Include units in your answer.
(c) Suppose that the solid from the previous parts is the interior of a vial and that the vial is filled with a salt water solution. The salt concentration at the surface (y = 6 cm) is 3 grams per Liter and at the tip (y = 1 cm) is 35 grams per Liter. Given that the concentration of salt in the water varies linearly as a function ρ(y) of y, write down a formula for ρ(y).
(d) Determine the amount of salt in the representative slice at height y and use your result to write down an integral the value of which is equal to the amount of salt in the solution. Finally evaluate the integral to get an approximate value for the amount of salt in the water and include units in your answers.
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