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FM Volumes of revolution II name _______________________
Objective Deadlines / Progress
Volu
mes
aga
inst
x-a
xis a
nd y
-axi
s
Apply the formulas for volumes of revolution against the x-axis and y-axis
Apply volumes of revolution to complex functions including trig functions, exponential and natural logarithm functions Know some standard volumes of revolution such as volume of a sphere or cone
Apply integration methods such as substitution and integration by parts
Volu
mes
with
par
amet
ric
equa
tions
Find areas under a curve given as parametric functions
Find volumes under a curve given as parametric functions against both xAxis and yAxis
Mod
ellin
g
Solve problems in context including using scaled models
FM Volumes of revolution II 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
Prior knowledge: You should to be able to find integrals using substitution methods; trigonome4tric identities and integration by parts
WB A1 The region R is bounded by the curve y=ex, the x-axis and the vertical lines x=0 and x=2Find the volume of the solid formed when the region is rotated 2π radians about the x-axis. Give an exact answer
WB A2 The region R is bounded by the curve y=e2 x, the x-axis and the vertical lines x = 0 and x = 4. 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 II name _______________________
WB A3 The region R is bounded by the curve y=sec x, the x-axis and the vertical lines x=π6 and
x=π3
Find the volume of the solid formed when the region is rotated 2π radians about the x-axis. Give an exact answer
WB A4 The region R is bounded by the curve y=sin2 x, the x-axis and the vertical lines x = 0 and x = π/2 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 π2
FM Volumes of revolution II name _______________________
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=4 ln x−1, the y-axis, x-axis and the horizontal lines y = 0 and y = 4 Show that the volume of the solid formed when the region is rotated 2π radians about the y-axis is 2π √e (e2−1 )
FM Volumes of revolution II name _______________________
WB B2 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
WB B3 The area bounded by the curve y=x2 and the lines x=3 and y=1 is rotated 2π about the line y=1 Find the volume of the solid formed
Can you generalise to give a formula for the volume formed when the curve is rotated about line y=a
FM Volumes of revolution II name _______________________
Volumes of revolution and Parametric Functions
Notes
When a curve is given in parametric equations we can use the following
Area=∫ y dx=∫ y dxdtdt
The volume of revolution formed when the parametric curve is rotated around the x-axis is given by
Volume=π ∫x=a
x=b
y2dx=π ∫t=q
t= p
y (t)2 dxdtdt
The corresponding volume of revolution formed when the parametric curve is rotated around the y-axis is given by
Volume=π ∫y=a
y=b
x2dy=π ∫t=q
t= p
x (t )2 dydtdt
Make sure you find the new bounds, p and q, of the integral
Areas with parametric functions
WB C1 Find the area under the curve given by the parametric equations:
x=2 t+1 y=t3−1t1≤t ≤2
WB C2 Find the area under the curve given by the parametric equations:
FM Volumes of revolution II name _______________________
x=sin θ y=cosθ 0≤θ≤ π2
Volumes with parametric functions
WB C3 The curve C has parametric equations:
x=t (1+t ), y= 11+t , t ≥0
The region R is bounded by C, the x-axis and the lines x = 0 and x = 2. Find the volume of the solid formed when R is rotated 2π radians about the x-axis.
FM Volumes of revolution II name _______________________
WB C4 Find the volume of revolution formed by rotating the curve x=cos t, y=√sin t−1 ,
0≤ t ≤ π2 by 2 around the x-axis
WB C5 Find the volume of revolution formed by rotating the curve x=e t, y=√t−1 , 2≤t ≤2π about the x axis
FM Volumes of revolution II name _______________________
WB C6 Find the exact volume of revolution formed by rotating the curve x=√ t−2, y=t2 , 2≤t ≤3 about the yAxis
WB C7 Find the exact volume of revolution formed by rotating the curve x=sin t , y=2t ,
FM Volumes of revolution II name _______________________
0≤ t ≤ π3 About the yAxis. Give your answer in terms of π
FM Volumes of revolution II name _______________________
Volumes of revolution and modelling
WB D1 The diagram shows a model of a goldfish bowl. The cross-section of the model is described by the curve with parametric equations
x=2sin t , and y=2cos t π6≤ t ≤ 11 π
6Where the units of x and y are given in cm. The goldfish bowl volume is formed by rotating the curve around the y-axis to form a solid of revolution.a) Find the volume of the water required to fill the model to a height of 3 cmb) The real goldfish bowl has a maximum diameter of 48 cm. Find the volume of water needed to fill the real bowl to a corresponding height.
4 cm
3 cm
FM Volumes of revolution II name _______________________
WB D2 The diagram shows the image of a gold pendant which has height 2 cm. the pendant is modelled by a solid of revolution of a curve C about the y-axis. The curve
has parametric equations x=cos t+ 12sin 2 t and y=−(1+sin t ) ,0≤t ≤2π
a) Show that a Cartesian equation of the curve C is x2=−( y 4+2 y3 ) (4)b) Hence, using the model, find in cm3, the volume of the pendant (4)