Integration using partial fractions

The volume, V microlitres, of a plant cell t hours after the plant is watered is modelled by the differential equation

The plant cell has an initial volume of 20 microlitres.

Find, according to the model, the time taken, in minutes, for the volume of the plant cell to reach 24 microlitres.

Show that

where A, B and k are constants to be found.

The model predicts that there is an upper limit, L microlitres, on the volume of the plant cell.

Find the value of L, giving a reason for your answer.

Express f (x) in partial fractions in terms of k.

Hence find the exact value of k for which

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

Find the first three terms, in ascending powers of x, of the binomial expansion of

writing each term in simplest form.

Using the answer to part (a) and using algebraic integration, estimate the value of

giving your answer to 4 significant figures.

Find, using algebraic integration, the exact value of

giving your answer in the form alnb + c, where a, b and c are constants to be found.

When chemical A and chemical B are mixed, oxygen is produced. A scientist mixed these two chemicals and measured the total volume of oxygen produced over a period of time. The total volume of oxygen produced, Vm³, t hours after the chemicals were mixed, is modelled by the differential equation

where k is a constant. Given that exactly 2 hours after the chemicals were mixed, a total volume of 3 m³ of oxygen had been produced,

solve the differential equation to show that

The scientist noticed that

• there was a time delay between the chemicals being mixed and oxygen being produced

• there was a limit to the total volume of oxygen produced

Deduce from the model

the time delay giving your answer in minutes,

the limit giving your answer in m³

Use the substitution x = u² + 1 to show that

where p and q are positive constants to be found.

Hence, using algebraic integration, show that

where a is a rational constant to be found.

Given that

find the values of the constants A, B and C

Hence, using algebraic integration, find the exact value of

giving your answer in the form a + bln 2 where a and b are integers to be found.

Explain why you can deduce that q = 4

Figure 4 shows a sketch of part of the curve C. The region R, shown shaded in Figure 4, is bounded by the curve C, the x-axis and the line with equation x = 3

Show that the exact value of the area of R is a ln 2 + b ln 3, where a and b are rational constants to be found.

Find, in its simplest form,

Hence find

Use integration by parts to find

giving your answer in the form a + b ln 2, where a and b are constants to be found.

Figure 1 shows a sketch of part of the curve with equation y x The finite region R, shown shaded in Figure 1, is bounded by the curve, the y-axis, the x-axis and the line with equation x = 1 The table below shows corresponding values of x and y for y =

Complete the table above by giving the missing value of y to 5 decimal places.

Use the trapezium rule, with all the values of y in the completed table, to find an estimate for the area of R, giving your answer to 4 decimal places.

Use the substitution u = e^x to show that the area of R can be given by

where a and b are constants to be determined.

Hence use calculus to find the exact area of R. [ Solutions based entirely on graphical or numerical methods are not acceptable. ]

Given that y 0, find

Use the substitution x = 4sin² to show that

Hence use integration to find

giving your answer in the form a + b, where a and b are exact constants.

A team of biologists is studying a population of a particular species of animal. The population is modelled by the differential equation

where P is the population in thousands, and t is the time measured in years since the start of the study. Given that P = 3 when t = 0,

solve this differential equation to show that

find the time taken for the population to reach 4000 for the first time. Give your answer in years to 3 significant figures.

The finite region R is bounded by the curve C, the x- axis, the line with equation x = 1 and the line with equation x = 4 This region is shown shaded in Figure 2 The region R is rotated through 360 ° about the x- axis.

Use calculus to find the exact volume of the solid of revolution generated, giving your answer in the form a + b ln c, where a, b and c are constants.

Use the substitution x = u², u > 0, to show that

Hence show that

where a and b are integers to be determined.

Find the values of the constants A, B and C.

where a and b are constants.

Find the particular solution of the differential equation

Hence obtain the solution of

for which y = 0 at x =x giving your answer in the form sec² x = g(y).

The curve C has parametric equations

The finite region R between the curve C and the x-axis, bounded by the lines with equations x = ln 2 and x = ln 4, is shown shaded in Figure 3.

Show that the area of R is given by the integral

Hence find an exact value for this area.

Find a cartesian equation of the curve C, in the form y = f(x).

State the domain of values for x for this curve.

Liquid is pouring into a large vertical circular cylinder at a constant rate of 1600 cm³ s^–1 and is leaking out of a hole in the base, at a rate proportional to the square root of the height of the liquid already in the cylinder. The area of the circular cross section of the cylinder is 4000 cm².

Show that at time t seconds, the height h cm of liquid in the cylinder satisfies the differential equation

When h = 25, water is leaking out of the hole at 400 cm³ s^–1.

Show that k = 0.02

Separate the variables of the differential equation

to show that the time taken to fill the cylinder from empty to a height of 100 cm is given by

Hence find the time taken to fill the cylinder from empty to a height of 100 cm, giving your answer in minutes and seconds to the nearest second.