Further Integration (A Level only) (OCR A Level Maths: Pure)

A student is estimating the area bounded by the curve y=fx, the x-axis and the lines and .

The student intends to find the area of two rectangles of equal width in order to estimate the area as shown in the diagram below.

By drawing a sketch, show how the student’s estimate of the area can be improved while still using rectangles of equal width.

Find the integral

Use calculus to evaluate

Find an expression for y given that

Integrate

Use calculus to find the value of

Find an expression for y given that

Use a suitable substitution to find

Use calculus and the substitution  to show that

Given that

use calculus to find the exact value of

Find 

Use integration by parts to find, in terms of e, the exact value of

Show that

can be written in the form

Hence find

writing your answer as a single logarithm (plus a constant of integration).

The diagram below shows a sketch of the curves with equations

and

Find the x-coordinates of the intersections of the two graphs.

Show that the area of the shaded region labelled R is given by

Use calculus to find the area of the shaded region labelled R.

The diagram below shows a sketch of the curve with equation 

The point   lies on the curve.

The shaded rectangle shown has width  δx and height y.

Calculate

Find the integral

Find an expression for y given that

Use calculus to evaluate

giving your answer as a single logarithm.

Integrate

Use calculus to find the value of

Use a suitable substitution to find the following

Use the substitution  to show that

where c is the constant of integration.

Show that, for (where k is an integer),

Use calculus and your result from part (a) to show that

Use calculus to find the exact value of

Use calculus to show that

Use integration by parts to find

Show that

where c is the constant of integration.

Express

as partial fractions.

Hence, or otherwise, find

The diagram below shows a sketch of the curves with equations

  and

Find the x-coordinates of the points of intersection of the two graphs.

Use calculus to find the total shaded area enclosed by the two graphs.

The diagram below shows a sketch of the curve with equation  .

A point   lies on the curve.

The shaded rectangle shown has width  and height .

Calculate

Find an expression for y given that

Integrate

Find an expression for y given that

Integrate

Use calculus and the substitution  to find the exact value of

Find

Show that

where k is a constant, and c is the constant of integration.

Use calculus to find the exact value of

writing your answer in the form  a ln b , where a and b are rational numbers to be found.

Find

Find

Find the integral

The diagram below shows a sketch of the curves with equations

    and      

Show that the two curves intersect at the points (3, 4), (4, 3) and (5, 0).

By using the substitution  , show that

where c is the constant of integration.

Using calculus, and your results from parts (a) and (b), show that the total shaded area enclosed by the two curves in the diagram is equal to

Use integration by parts to show that

where c is the constant of integration.