5.4 Further Integration

Integrate

Show that

where  is a constant of integration.

Find an expression for  given that

and that    when  .

Find the indefinite integral for

Find the indefinite integral for

Given that ,  find  .

Hence, or otherwise, find

Consider the function  .

Find 

Hence, find

Let 

Find    given that  .

Show that

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

Point  is the point of intersection of the curve    with the ­ ­-axis.  Point   is the point of intersection of the curve  with the line for which .  Point  is the point of intersection of the line    with the -axis.

Write down the -coordinates of points

Calculate the area of the shaded region.

The diagram below shows the graph of the function  which is defined by

The shaded region in the diagram is the region enclosed by the -axis and the graph of .

Find the area of the part of the shaded region that lies above the -axis.

Find the area of the entire shaded region.

The diagram below depicts the design for a new company logo. The logo is formed by a circle centred on the origin, which is divided into two regions by the curve  where  is the function defined by . The points where the circle and the curve intersect lie on the -axis, as shown.

The shaded region in the diagram is the region inside that circle that lies below the curve  .

Find the percentage of the circular logo that is shaded.

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

(i)

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

(ii)

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 indefinite integral for

Find the indefinite integral for

Find an expression for given that

Find the indefinite integral

Find an expression for given that

and also that  when 

Find the indefinite integral for

Find the indefinite integral for

Find the indefinite integral

Let   for  

Find   given that  .

Show that

A curve with equation  is such that

where  is a real constant. 

Given that the curve passes through the points  and , find .

Explain why

Use definite integration, along with the result from part (a), to show that

Using your knowledge of the natural logarithm function, explain (without using your GDC) why the value of the integral found in part (b) is a positive number.

The diagram below shows the graph of the function which is defined by

The shaded region in the diagram is the region enclosed by the -axis and the graph of .  The three sub-parts of the shaded region are denoted by R, S and T, as shown.

Find the value of

Find the individual areas of each of the three sub-parts R, S and T of the shaded region.

Compare the sum of the answers in part (b) to the answer in part (a) and comment on the result.

The diagram below depicts the design for a new company logo.  The upper border of the logo is formed by a part of the curve with equation  ,  while the lower border of the logo is formed by a part of the curve with equation .  As shown in the diagram, the logo is divided into a shaded part and an unshaded part by a part of the curve with equation .

Find the percentage of the total area of the logo that is shaded.

The following diagram shows a part of the graph of the curve with equation ,  where  is a constant.  The point marked A is the vertex of the curve.  Region  is the region enclosed by the curve and the -axis, for the part of the curve where  is non-negative.  Region  is the region enclosed by the curve, the positive -axis, and the line through point A with gradient zero.

When region  is rotated  radians about the -axis, the resultant solid of revolution has a volume equal to . 

Find the area of region  by calculating an area between the curve and the -axis.

Find the area of region S by calculating an area between the curve and the -axis.  Confirm that this matches your answer to part (a).

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.