Spring 2005 MATH 172
Week in Review VI
courtesy of David J. Manuel
Section 9.2, 9.3, 9.6
Section 9.2
1. Derive the formula for the Integrating Factor of a First-Order, Linear Differential Equation.
2. A tank with a 500 gallon capacity originally contains 200 gallons of water with 100 lb of salt in
the solution. Water containing 1 lb of salt per gallon enters the tank at a rate of 3 gal/min, but the
mixture leaves the tank at a rate of only 2 gal/min. Find the amount of salt in the tank at any time
t before the tank begins to overflow. Include appropriate restrictions on t.
Section 9.3
3. Use the Mean Value Theorem to derive the formula for the length of the differentiable curve
x = g(y) from y = c to y = d.
4. Use the arclength formula to prove that the circumference of a circle of radius r is 2πr:
a) by integrating with respect to x or y
b) by parametrizing the circle and integrating with respect to t.
Section 9.6
6. A tank contains a fluid with weight density ρg. The side of the tank is in the shape of the region
bounded by x = h(y), 0 ≤ y ≤ c in the first quadrant. If the tank is full, write a Riemann Sum to
APPROXIMATE the force against one side of the plate.
7. A rectangular plate 3 ft wide and 5 ft tall is placed in a pool at an angle of θ from the vertical
(see figure below). If the top of the plate is 1 foot below the surface, write an integral to find the
hydrostatic force on the plate.
θ