FM Volumes of revolution I name _______________________

Objective Deadlines / Progress

Volu

mes

of r

evol

ution

Know the formula for volumes of revolution against the x-axis

Know the formula for volumes of revolution against the y-axis

Know some standard volumes of revolution such as for a sphere

Mod

ellin

g / p

robl

ems Add and subtract volumes to solve more

complex problems

Apply volumes of revolution to problems in context

FM Volumes of revolution I name _______________________

Volume of revolution around x-axis

Notes

The volume of revolution formed when y=f(x) is rotated around the x-axis between the x-axis, x=a and x=b is given by

Volume=π∫a

b

y2dx

WB A1 The region R is bounded by the curve y=2x+1 and the vertical lines x=1 and x=3 Find the volume of the solid formed when the region is rotated 2π radians about the x-axis.

FM Volumes of revolution I name _______________________

WB A2 The region R is bounded by the curve y=1x , the x-axis and the vertical lines x = 1 and x =2.

Find the volume of the solid formed when the region is rotated 2π radians about the x-axis. Write your answer as a multiple of π

WB A3 The region R is bounded by the curve y=x √ x, the x-axis and the vertical lines x = 0 and x =2. Find the volume of the solid formed when the region is rotated 2π radians about the x-axis. Write your answer as a multiple of π

FM Volumes of revolution I name _______________________

WB A4 The region R is bounded by the curve y=√3−x3, the x-axis and the vertical lines x = -1 and x = -3 Find the volume of the solid formed when the region is rotated 2π radians about the x-axis. Give your answer as an exact multiple of π

WB A5 The region R is bounded by the curve y=x (1−x), the x-axis and the vertical lines x = 0 and x =1 Find the volume of the solid formed when the region is rotated 2π radians about the x-axis.

WB A6 Find the volume of the sphere formed when a circle of radius r, centred at the origin is

FM Volumes of revolution I name _______________________

rotated 2π radians about the x-axis.

FM Volumes of revolution I name _______________________

Volumes of revolution around the y-axis

Notes

The volume of revolution formed when x=f(y) is rotated around the y-axis between the y-axis, y=a and y=b is given by

Volume=π∫a

b

x2dx

When you use this formula you are integrating with respect to y. So you may need to rearrange functions accordingly

WB B1 The region R is bounded by the curve y=√x−1, the y-axis and the horizontal lines y=1 and y=3Find the volume of the solid formed when the region is rotated 2π radians about the y-axis. Give your answer as a multiple of π

FM Volumes of revolution I name _______________________

WB B2 The region R is bounded by the curve x=√2 y−1, the y-axis and the vertical lines y=4 and y = 8Find the volume of the solid formed when the region is rotated 2π radians about the y-axis. Give your answer as a multiple of π

WB B3 The region R is bounded by the curve y=x2−2the y-axis and the vertical lines y=1 and y=3Find the volume of the solid formed when the region is rotated 2π radians about the y-axis. Give your answer as a multiple of π

FM Volumes of revolution I name _______________________

WB B4The region R is bounded by the curve y=1x the y-axis and the vertical lines y=1 and y=2

Find the volume of the solid formed when the region is rotated 2π radians about the y-axis. Give your answer as a multiple of π

WB B5 The region R is bounded by the curve y=√xthe y-axis and the vertical line y=2 Find the volume of the solid formed when the region is rotated 2π radians about the y-axis

FM Volumes of revolution I name _______________________

Adding and subtracting volumes

WB C1 The region R is bounded by the curve y=x3+2, both the x-axis and the y-axis and the line y=5−2 x a) Find points A and Bb) Find the volume of the solid formed when the region is rotated 2π radians about the x-axis. Give your answer as a multiple of π

WB C2The region R is bounded by the curves y=√x, y= 1

8 x and the line x=1

Find the volume of the solid formed when the region is rotated 2π radians about the x-axis. Give your answer as a multiple of π

R

FM Volumes of revolution I name _______________________

Volumes of revolution and Modelling

WB D1 A manufacturer wants to cast a prototype for a pen barrel out of solid resin. The region bounded by y=k−100 x2 and the x-axis as shown in the diagram is used as a model for the cross-section of the pen barrel. a) suggest a suitable value for kb) Use your value of k to estimate the volume of resin needed to make the prototypec) State one limitation of this model

FM Volumes of revolution I name _______________________

WB D2

Figure 1 shows the central cross-section of a bird bath made of concrete. Using the given measurements the cross sectional curve CD is modelled as a curve with equation y=1+k x2 −0.2≤x ≤0.2 as shown in figure 2

a) Suggest the maximum depth of the bird bath (1)b) Find the value of k (2)c) Hence, find the volume of concrete required to make the bird bath according to this model.

Give your answer to 3 sf (7)d) State a limitation of the model (1)e) It was later discovered that the volume of concrete used to make the bird bath was 0.127m3

Using this information, evaluate the model, explaining your reasoning (1)