Slide 1
Volume
Volume by SlicingUse when given an object where you know the
shape of the base and where perpendicular cross sections are all
the same, regular, planar geometric shape.
Volume of a Solid of known integrable cross section - area A(x)
from a to b, defined as
Procedure: volume by slicing sketch the solid and a typical
cross section find a formula for the area, A(x), of the cross
section find limits of integration integrate A(x) to get volume
ExampleFind the volume of a solid whose base is the circle and
where cross sections perpendicular to the x-axis are all squares
whose sides lie on the base of the circle.
First, find the length of a side of the squarethe distance from
the curve to the x-axis is half the length of the side of the
square solve for y
length of a side is :
ExampleA solid has a circular base. Find the volume of the solid
if every plane cross section is an equilateral triangle,
perpendicular to the x-axis, given the equation of the circle to
be
Solution
length of a side:height of triangle found using Pythagorean
Theorem:Area:Volume:
Volume of a Solid of RevolutionA solid of revolution may be
generated by revolving the area under the graph of a continuous,
non-negative function y = f(x) from a to b, about the x-axisCommon
Examplescone generated by revolving the area under a line that
passes through the origin, from 0 to k, about the x-axis
cylinder generated by revolving the area under a horizontal line
of positive height from a to b, about the x-axis
ExampleLook at the region between the curve and the x-axis, from
x = 0 to x = 2, and revolve it about the x-axis
If we slice the resulting solid perpendicular to the x-axis,
each cross section is a circle, or diskThe radius of the circle is
the distance from the curve to the axis of rotationArea of a cross
section (circle)
The thickness of each disk is infinitely small, so it is
represented as dx. By adding the area of all these circles, from
x=0 to x=2, we would get the volume of the solid.
Disk Method
radius is always perpendicular to the rotational axisrotational
solid with no hollow parts will always have a circular cross
section, with area use as long as there is NO hollow
spaceFormula:
Example: Disk MethodRotate the area bounded by and the x-axis,
about the x-axis; then calculate the volume
Calculus AppletVolumes of Revolution
circular cross section
Solution
Find limits of integration Find radius of cross section
Volume
Example: Disk MethodFind the volume generated by rotating the
area in the first quadrant bounded by the curve, the y-axis, and
the line y = 9, around the y-axis.
Use y as variable of integrationCalc Applets Volumes of
Revolution
circular cross section
Solution
limitsfrom y = 0 to y = 9 radius Answer
Washer Methodused when solid has hollow partsradius of rotation
perpendicular to the axis of rotationtwo radii - outer and inner
FormulaR outer radius r inner radius
Example: Washer MethodFind the volume of the solid when is
rotated about the line y = -2, from x=0 to x=2
outer radius R
inner radius r
Solution
Find outer and inner radiiouter - curve furthest away from the
axis of rotation; subtract from this, the axis of rotation
inner - inside curve, forms the hollow wall of the solid
Example: Washer MethodFind the volume of the solid of revolution
generated by revolving about the y-axis, the area enclosed by the
graphs of
outer radius R
inner radius r
Solution
solve for limits
to revolve about the y-axis, need to solve each equation for
x
radii - in relation to the y-axis, the outer radius is and the
inner radius is
Example: Washer MethodConsider the area captured between the
graphs of
What volume is generated if this area is rotated about the
x-axis ?
limits
radii
answer
Example: Washer MethodFind the volume of the solid generated
when the region bounded by
is rotated about the line y = -3
limitsOnly way is to obtain from graphing calculator
radii
Answer
