5.9 Advanced Integration

Find the following indefinite integrals:

Using a sketch, briefly describe the family of graphs corresponding to all the possible specific solutions to the integral in part (a).

Show that

    =  

where A and B are constants to be found.

Hence, find the indefinite integral

     
 

using laws of logarithms to simplify your answer as far as possible.

Show that

. 

Use the substitution to find the following indefinite integral:

 .

Hence find the value of the definite integral

     

by evaluating the definite integral entirely in terms of  

by converting the integral limits to appropriate values of  and evaluating the definite integral entirely in terms of .

Verify that the two methods give the same result for the value of the integral.

Show that   may be written in the form , where  and  are constants to be determined.

Using your results from part (a) along with the substitution , show that 

  .

Find the exact value of the definite integral

Use the substitution to show that 

   .

Use integration by parts to find the indefinite integral

.

Hence find the exact value of the definite integral

 .

Use technology to evaluate the integral in part (b), and compare this to the exact value you found.

Use integration by parts twice to show that

 

where ,  and  are constants to be found, and where  is a constant of integration.

Let  be a function defined for all .  Consider the graph of  .

Given that 

and that the graph passes through the point ,  find an expression for .

Let  be the function defined by  .

The diagram below shows a part of the graph of the curve  .  The shaded region is the region bounded by the curve, the positive -axis and the line .

Find the exact area of the shaded region.

The following diagram shows a part of the graph of the curve  .  The shaded region is the region enclosed by the graph and the positive - and -axes.

For the part of the curve that forms the boundary of the shaded region, show that  .

Find the area of the shaded region

by calculating it as an area between the curve and the -axis.

by calculating it as an area between the curve and the -axis.

Find the volume of the solid formed when the shaded region is rotated radians about the -axis.

Find the volume of the solid formed when the shaded region is rotated radians about the -axis.

The diagram below shows the cross-section of a bowl that a company is planning to begin producing.

As indicated on the diagram, one of the sides of the bowl in the cross-section may be described by the curve ,  where units for  and  are centimetres.  The cross-section is entirely symmetrical about the -axis.  The flat circular bottom of the bowl has a diameter of 12 cm, and the vertical depth of the bowl is 6 cm.  For purposes of answering this question, the thickness of the bottom and sides of the bowl may be regarded as negligible.

Find the exact coordinates of the point marked  on the diagram.

Show that the capacity of the bowl in cm³ is given by 

 where  is a constant to be determined.

Hence find the capacity of the bowl.

Find the following indefinite integrals: 

Using a sketch, briefly describe the family of graphs corresponding to all the possible specific solutions to the integral in part (a).

Show that

Use the substitution  to find the exact value of the following definite integral:

being sure to simplify your answer as much as possible.

Find the exact value of the definite integral

Use the substitution  to find the indefinite integral

Use integration by parts to find the indefinite integral

Hence find the exact value of the definite integral

Show that

where  are non-zero constants, and c is a constant of integration.

A continuous random variable has the probability density function  given by

Find the value of .

Find the indefinite integral

Let be the function defined by . 

Show that  is an even function.

The diagram shows a part of the graph of .  The shaded region is the region bounded by the curve, the positive -axis, the negative -axis, and the line .

Find the exact area of the shaded region.

Given that

write down the value of .  Justify your answer.

Consider the curve with equation

where , .  It is given that, for any allowed value of  has a defined non-negative value for all values of  in the stated domain. 

Let  be the area enclosed by the curve, the -axis, and the lines    and  .

 Find an expression for in terms of .

The diagram below shows the cross-section of a goldfish bowl to be produced by Pieseize Manufacturing, a specialist company supplying products for goldfish enthusiasts.

The glass part of the bowl sits on a solid base, indicated by the shaded region on the diagram. The cross-section of the glass part of the bowl is symmetrical about the -axis, and may be described by the curve with equation

The dashed horizontal line represents the diameter of the open top of the fishbowl.  The maximum depth of the fishbowl, measured along the -axis from the diameter of the open top to where the glass part of the bowl meets the base, is indicated by  in the diagram.  All coordinates are expressed in centimetres, and for purposes of answering this question the thickness of the glass sides of the bowl may be regarded as negligible.

The owner of the company, Skodyn Pieseize, is extremely superstitious and is obsessed with the number 23. Therefore he insists that the capacity of the glass part of this new fishbowl must be exactly 23 litres.  Find the value of  that satisfies this requirement.