5.4 Further Integration

Integrate

Find the definite integral

Find an expression for y given that

Using a suitable substitution, show that

Given that

use calculus to find the exact value of

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 the graphs of the line and  the curve 

Work out the x-coordinates of the points labelled P, Q and R.

Work out the area of the shaded region.

Consider the function such that

Find

Find

Find

Consider the function  .

Find 

Hence, find

Let  

Find given that  .

Find the indefinite integral for

Find the indefinite integral for

Find an expression for y given that

Find the exact value of

Find the definite integral

Find an expression for y given that

and also that  when 

Use a suitable substitution to show that

Let I be the definite integral defined by

where a, b and k  are real constants such that  and .

Show that

Hence find the exact values of 

Explain why

Use the result from part (a) to show that

Explain why the value of the integral found in part (b) is a positive number.

The diagram below shows a sketch of part of the curves with equations    and  .

The shaded region in the diagram is the area bounded by the two curves.

By first showing that the area of the shaded region is given by

calculate the exact area of the shaded region

Explain why your answer to part (a) is not affected by the fact that the shaded region is partially above and partially below the x-axis.

The diagram shows a sketch of part of the curves with equations  and    where a, b, c and d are constants with  .

The x-coordinates of the points of intersection of the two curves are p, q and r, where    Region S is the region enclosed by the two curves between    and ,  while region T is the region enclosed by the two curves between    and 

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

The curve intersects the x-axis at the points and, and region U is the region enclosed by the curve and the x-axis between    and .

Given that the areas of regions S and U are equal, calculate the total area enclosed by the two curves in the first diagram.  Be sure to provide a suitable justification for your answer.

Find

Find

Given that   and ,  find

Show that 

A function f is defined by 

Let I be the definite integral defined by

where    is a constant.

Determine the value of I, giving your answer in terms of a.

Hence, or otherwise, determine the value of a which maximises the value of I, and calculate the value of I when a takes that value.