Solve the following differential equations using the substitutions given. (a) (b)

(a) The figure on the right shows the ellipse and the line x = 1. By using the substitution , find the area of the shaded region, giving your answer in an exact form. (b) Find the exact volume generated when the shaded region is rotated through 2π radians about the y-axis.

A rectangular tank with its base horizontal is filled with water to a depth h at time t = 0. Water leaks out of the tank from a small hole in the base at a rate proportional to the square root of the depth of the water. If the depth of water is ½h at time T, find […]

A researcher is investigating the spread of a certain disease in a town with a population of 3000 people. The researcher suggests that I, the number of people infected by the disease at time t days satisfies the differential equation , where k is a positive constant. (i) Given that I = 30 when t = 0, […]

The diagram shows a river,500m wide between the straight parallel banks where AD = 500m and DB = 2000. A man swims across the river from A to C at 0.5 m/s and then he runs along the banks from C to B at 0.8 m/s (a)If he swims in the direction AC making an […]