Further Integration (CIE A Level Maths: Pure 1)

The diagram below shows part of the graph of 

Write down the values of where

Show that
      

Evaluate

Write down the area of the region labelled .

The diagram below shows part of the graph of .

Find the area of the shaded region labelled .

Find the -coordinates of the intercepts of the line with equation and the curve with equation 

Evaluate
      

The diagram below shows the graphs of and and
Find the exact area of the shaded region .

A student is estimating the area bounded by the curve the  -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.

The diagram below shows the graphs of the line  and  the curve .

Work out the  -coordinates of the points labelled .

Work out the area of the shaded region.

The diagram below shows a sketch of the curves with equations

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

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

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

The diagram below shows the graph of the curve with equation 

Find the -coordinates of the points where  the graph of intercepts the -axis.

The shaded region, , is to be rotated around the  -axis.
Find the volume of the shape generated.

The diagram below shows the graphs of two horizontal lines and the function  

where are constants.

Point  has coordinates .

Point  has coordinates .

Find the values of .

The shaded region, , is to be rotated  around the -axis.
Find the volume of the shape generated.

The diagram below shows part of the graph of .

Find the area of the shaded region labelled .

The diagram below shows part of the graph of

Find the total area of the two shaded regions.

The diagram below shows a sketch of the curve with equation 

The point   lies on the curve.

The shaded rectangle shown has width  and height . 

Calculate

The line with equation  cuts the curve with equation  at the points  and Q, as shown.

Find the - and -coordinates of the points and .

Find the exact area of the region labelled , giving your answer in the form
where  and are integers to be found.

The diagram below shows the graphs 

Find the exact area of the shaded region.

The diagram below shows a sketch of the curves with equations

and 

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

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

A mathematical model for a bowl is obtained by rotating through  about the -axis the part of the curve   which is between  and   and then adding a flat bottom.   is a constant such that .

Find the capacity of the bowl in terms of .

In the case where the boundaries   and  are used in place of  and ,  the capacity of the bowl is increased by cubic units.
Find the value of .

What assumption has been made in finding the capacity of the bowl?

The diagram below shows a right-angled triangle with vertices at the origin, the point and the point ,  where and .

Find an equation of the line on which the hypotenuse of the right-angled triangle lies, giving your answer in the form .

The triangle is rotated  about the -axis to form a cone.
Thus use calculus to prove that the general formula for the volume , of a cone is

where  is the base radius of the cone and  is its perpendicular height.

The diagram below shows part of the curve defined by the equation  where is a positive constant.  The shaded region  is bounded by the curve, the  -axis, and the lines  and  

Given that the volume of the solid formed when the region   is rotated  about the -axis is  cubic units, find the value of .

The diagram below shows part of the graph of .

Find the area of the shaded region labelled .

Without doing any additional calculation, explain why  must be equal to zero.

The diagram below shows part of the graph of

Find the total area of the two shaded regions.

The diagram below shows the graphs of    and .

Find the exact area of the shaded region.

The diagram below shows the graphs of   and

Find the total area of the shaded regions.

The diagram below shows the graphs of  

Find the area of the shaded region, .

Sketch the region described by the following inequalities

The region described in part (a) is rotated through about the -axis.
Find the volume of the solid formed, giving your answer in terms of .

The solid formed in part (b) will have a ‘hole’ in its centre.

Find the volume of this ‘hole’, giving your answer in terms of  .

Hence show that there are no values of  in the given interval that make the volume of the solid equal to the volume of the ‘hole’.

Starting with the equation of a semicircle of radius ,  (where  ),  use calculus to prove that the general formula for the volume,, of a sphere of radius is

      .

The diagram below shows a sketch of the curves with equations

The region  is bounded by the -axis and the two curves.
This region is rotated through about the -axis.
Find the volume of the solid formed.

The regions marked  are now to be considered as a single region bounded by the two curves.

A student has correctly constructed an integral to calculate the volume formed by rotating that combined region through  about the -axis, and has correctly calculated its value,.

How does  compare to the answer you found in part (a)?

Be sure to fully explain your answer.