Exam 8: Applications of Integration

The equations x = 2, x = 4, y = 1/x, and y = 0 define the bounds of a region of the plane. Find the volume of the solid obtained by rotating the region about the x-axis.

Given that the area of a sphere of radius r is 4 $\pi$ , find the centroid of the semicircular arc. .

A tank is in the shape of a right circular cone with top radius 2 m and height 5 m. It contains water up to the height of 4 m. The density of water is 1,000 kg/m³. How much work must be done to pump all of the water out of the tank over the top edge of the tank?

A conical tank with vertex at the bottom and top radius 10 cm is 40 cm tall. It is filled with a substance whose density at depth y cm is (2 + y) g/ . Find the total mass of the substance filling the tank.

Find the centroid of the planar region bounded by y = , y = 0, x = 1, and x = 2.

Find a general solution to the first-order linear differential equation - 2y - = 0.

Find the function satisfying (x) = x , F(0) = 1.

Find the centroid of the finite plane region bounded by y = , y = , and x = 1.

Determine the solution of the differential equation = (1 - 2x) that satisfies the condition y(0) = - .

An experiment has six possible outcomes with values X from 0 to 5 inclusive. The probability of the outcome with value j is proportional to . Find the mean and standard deviation of the distribution of values X.

A motorboat is travelling with a velocity of 40 ft/sec when its motor shuts off at time t = 0. Thereafter its deceleration, due to water resistance, is given by = -k where k is a positive constant. After 10 seconds, the boat's velocity is measured to be 20 ft/sec. (a) What is the value of k? (b) When will the boat slow to 5 ft/sec?

Find the general solution of the differential equation + t y = (t + 1).

A tank contains 200 litres of fluid in which 60 grams of salt is dissolved. A brine solution of 2 grams of salt per litre is pumped into the tank at a rate of 2 litres/minute. The well-stirred solution is then pumped out at the same rate.(a) Write the differential equation in terms of Q(t) and t and write the initial condition to model this problem. (Q(t) should represent the amount of salt in the tank at time t).(b) Solve the equation in part (a).(c) How much salt will be present in the tank after a long period of time?

An event A and its complement are disjoint.

Find a general solution to the first-order linear differential equation x + y = x³.

Find the moment about the x-axis of a thin plate that occupies the planar region described by 0 $\le$ y $\le$ , 0 $\le$ x $\le$ 1 if the areal density is given by (x) = e^x.

Find the coordinates of the centroid of the region enclosed by y = sin(x) and the x-axis from x = 0 to .

Find the volume of a right circular cone of base radius r and height h.

A cylindrical tank 8 m in diameter and 8 m high, with axis vertical, has 6 m of water in it. How many kilogram metres of work must be done to pump half the water in the tank to a height of 8 m above the top of the tank? (1 m³ of water weighs 1,000 kg.)