ch 7 c volumes
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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