Applications of Integration (Edexcel IGCSE Further Pure Maths): Exam Questions

Show that

Hence express in the form where and are integers, giving the value of and the value of ,

(i) Find

(ii) Hence evaluate , giving your answer in the form where , and are integers.

Given that can be written in the form where and are constants,

find the value of , the value of and the value of .

(i) Hence write down the maximum value of ,

(ii) Write down the value of for which this maximum occurs.

The curve has equation

The line with equation intersects at two points.

Find the coordinates of these two points.

The finite region bounded by the curve and the line is rotated about the -axis.

Use algebraic integration to find, to 3 significant figures, the volume of the solid generated.

Figure 3 shows part of the curve with equation and part of the curve with equation

The curve and the curve intersect at the point

Find the coordinates of point

The finite region , shown shaded in Figure 3, bounded by the curve , the curve and the straight line is rotated through 360º about the -axis.

Find, using algebraic integration, the exact volume of the solid formed.

Figure 3 shows part of the curve with equation and part of the line with equation

The region , bounded by the -axis, the curve and the line , is rotated through 360° about the -axis.

Using algebraic integration, find the exact value of the volume of the solid generated.

Using a formula (opens in a new tab), show that

Using a formula (opens in a new tab), show that

Hence show that

where , and are integers to be found.

Figure 3 shows part of the curve with equation and part of the curve with equation

Point is the intersection of and as shown in Figure 3

Point is the intersection of with the -axis as shown in Figure 3

Point is the intersection of with the -axis as shown in Figure 3

The finite region , shown shaded in Figure 3, is bounded by the -axis, and

Use calculus to find, in its simplest form, the exact area of

Given that can be written in the form where , and are constants,

find the value of , the value of and the value of

Given that can be written in the form where , and are constants,

Hence, or otherwise, find

(i) the value of for which has its greatest value

(ii) the greatest value of

The curve has equation

The curve with equation intersects curve at two points.

Find the coordinate of each of these two points.

Given that can be written in the form where , and are constants,

The curve has equation

The curve with equation intersects curve at two points.

Use algebraic integration to find the exact area of the finite region bounded by the curve and the curve

The curve with equation where and the line with equation where intersect at the points and , as shown in Figure 2.

(i) Show that the coordinates of point are (0, 2)

(ii) Find the coordinates of the point

The finite region bounded by and , shown shaded in Figure 2, is rotated through radians about the -axis.

Use algebraic integration to find the volume of the solid generated.
Give your answer in terms of

Expand in ascending powers of up to and including the term in

Where appropriate express each coefficient as an exact fraction in its lowest terms.

Write down the range of values of for which your expression is valid.

Express in the form where and are rational numbers whose values should be stated.

Obtain a series expansion for in ascending powers of up to and including the term in

Hence, using algebraic integration, obtain an estimate of

Give your answer to 5 significant figures.

Figure 2 shows the graph of part of the curve with equation
The finite region enclosed by the curve and the straight line with equation is rotated through 360° about the -axis.

Use algebraic integration to find the exact volume of the solid generated.
Give your answer in terms of

Expand in ascending powers of , up to and including the term in giving each coefficient as an integer.

where and are prime numbers

Given that the fourth and fifth terms, in ascending powers of , in the series expansion of are 20and 48 respectively,

find the value of and the value of

Using the first five terms, in ascending powers of , in the series expansion of

obtain an estimate, to 4 significant figures, of

The region , shown shaded in Figure 2, is bounded by the curve with equation and the curve with equation

The two curves intersect at the point and at the point .

Find the coordinate of the point and the coordinate of the point .

The region is rotated through 360° about the ‑axis.

Use algebraic integration to find the volume, to 2 decimal places, of the solid generated.

Figure 3 shows part of the curve with equation
The curve crosses the -axis at the point and the -axis at the point .

(i) Write down the coordinate of point .

(ii) Show that the coordinate of is

The line is the normal to at the point .

Find an equation for , giving your answer in the form

The finite region is bounded by , and the -axis.

Using calculus, find the area of .
Give your answer to one decimal place.