Areas and Volumes of Revolution 2024 Final

Stuff to do with integration :)

1. Integration Areas and Volumes of Revolution Specialist Mathematics Unit 4, Topic 1

2.   Finding Areas You’re already familiar with the idea that definite integration gives you the (signed) area bound between the curve and the -axis from Methods. Given your expanded integration skills, you can now find the area under a greater variety of curves.   The diagram shows part of the curve The region is bounded by the curve, the -axis and the lines and , as shown in the diagram. Use integration to find the area of .             Recall that when we integrate , we divide both by the increased power , but also by .   ?

3.     Area between two curves             The areas under the two curves are and . It therefore follows the area between them (provided the curves don’t overlap) is:   Fro Tip : Ensure you have top curve minus bottom curve. [Textbook] The diagram shows part of the curves and where . The region is bounded by the two curves. Use integration to find the area of .             ?

4. Test Your Understanding An exam question… ?

5. You should be comfortable with this application as you have also covered it in Methods in Unit 3. Now let’s move onto a new application, solids of volumes of revolutions…

6. Area under a graph gives the area bounded between , , and the -axis. Why?         If we split up the area into thin rectangular strips, each with width and each with height the for that particular value of . Each has area .   If we had ‘discrete’ strips, the total area would be: But because the strips are infinitely small and we have to think continuously, we use instead of . Integration therefore can be thought of as a continuous version of summation.          

7. Volumes of Revolution Now suppose we spun the line about the axis to form a solid (known as a volume of revolution ):   Click to Start animation How could we use the same principle as before to express the volume? What should we use instead of strips?     We’re summing a bunch of infinitely thin cylinders , each of width and radius . Each has a volume of: Thus the volume is: ! i.e. Square the function and slap a on front.   Reveal        

8. Example of this practice [Textbook] The diagram shows the region which is bounded by the -axis, the -axis and the curve with equation . The region is rotated through about the -axis. Find the exact volume of the solid generated.           Find roots first: Volume:  

9. Volumes of Revolution with harder integration Recap : When revolving around the -axis,   [Textbook] The region is bounded by the curve with equation , the -axis and . Find the volume of the solid formed when region is rotated through radians about the -axis.       ?

10. Test Your Understanding The finite region R which is bounded by the curve C , the x -axis and the line x = 125 is shown shaded in Figure 3. This region is rotated through about the x -axis to form a solid of revolution.   Use calculus to find the exact value of the volume of the solid of revolution. (5)   Past exam question… ?

11. Revolving around the -axis               If the curve is revolved around -axis:   To revolve instead around the -axis, we simply swap the roles of the and axes!   The diagram shows the curve with equation . The region is bounded by the curve, the -axis and the lines and . The region is rotated through about the -axis. Find the volume of the solid generated.                

12. Revolving around the -axis: your turn!   Recap : When revolving around the -axis, i.e. we are just swapping the roles of and .   The diagram shows the curve with equation . The finite region , shown in the diagram, is bounded by the curve, the -axis, the -axis and the line . Region is rotated by radians about the -axis. Use integration to show that the exact value of the volume of the solid generated is .           Since we’re finding , we need to find in terms of .   ?

13. Adding and Subtracting Volumes     Suppose we wanted to revolve the following area around the -axis. What strategy might we use to find the volume of this resulting solid? Find the volume of revolution for the top curve. Then cut out (subtract) the cone.     This bit is a cone. Volume Reminders:            

14. Example: look at the procedure! The region is bounded by the curve with equation , the line and and -axes. Verify that the coordinates of are . A solid is created by rotating the region about the -axis. (b) Find the volume of this solid.                    Find the two volumes separately: intersects the -axis at 2.5                       It’s a cone!

15. Volumes by Subtraction, your turn! The diagram shows the region bounded by the curves with equations and and the line . The region is rotated through about the -axis. Find the exact volume of the solid generated.               Do volume under top curve and subtract volume under bottom curve. Point of intersection:   ?

16. Test Your Understanding Past exam question… ? ?

17. Volumes of revolution for parametric curves If this was to be asked, it would be complex unfamiliar! Look at the notation used. We know that parametric equations are where, instead of some single equation relating and , we have an equation for each of and in terms of some parameter, e.g. . As varies, this generates different points . To integrate parametrically, the trick was to replace with       Note that as we’re integrating with respect to now, we need to find the equivalent limits for . We can do the same for revolving around the -axis: just replace with and change the limits.   The curve has parametric equations , , . The region is bounded by , the -axis and the lines and . Find the exact volume of the solid formed when is rotated radians about the -axis.   When When We have partial fractions:  

18. Test Your Understanding Past exam question… When , When ,   ?

19. Modelling with Volumes of Revolution A manufacturer wants to cast a prototype for a new design for a pen barrel out of solid resin. The shaded region shown in the diagram is used as a model for the cross-section of the pen barrel. The region is bounded by the -axis and the curve with equation , and will be rotated around the -axis. Each unit on the coordinate axes represents 1cm. (a) Suggest a suitable value for . (Let’s say pens are 10cm long) (b) Use your value of to estimate the volume of resin needed to make the prototype. (c) State one limitation of this model.         (pens are around 10-15cm long)   The cross-section of the pen is unlikely to match the curve exactly. a b c ? ? ?

20. Modelling with Volumes of Revolution The diagram shows a model of a goldfish bowl. The cross-section of the model is described by the curve with parametric equations , , where the units of and are in cm. The goldfish bowl is formed by rotating this curve about the -axis to form a solid of revolution. Find the volume of water required to fill the model to a height of 3cm. The real goldfish bowl has a maximum diameter of 48cm. (b) Find the volume of water required to fill the real goldfish bowl to the corresponding height.   3cm 4cm We’re revolving around the -axis, so use When When   Linear scale factor = 12 Volume scale factor = Volume in actual tank cm 3 (3sf)   a b This is a well-established strategy for integrating powers of or   Can use integration by inspection (‘reverse chain rule’). This is one to remember. ? The and corresponds to the two sides of the bowl. But we only want to integrate the curve on one side of the bowl, so choose one.   ?