(SEC. 7 .3 DAY ONE)
Volumes of Revolution
DISK METHOD
Example 1: Find the volume of a sphere with a radius of 2.
METHOD 1: GEOMETRY!!!!!!!!!!
๐=43
๐ ๐3
๐=43
๐ (2)3
๐=323
๐
๐ โ33.51๐ข๐๐๐ก๐ 3
Example 1: Find the volume of a sphere with a radius of 2.
METHOD 2: CALCULUS!!!!!!!!!!Step One: Write an equation to represent the edge of the shape (in this case: a CIRCLE!!!).
Step Two: Solve the equation for y.
What shape would result from rotating this โfunctionโ over the x-axis?
๐๐+๐๐=๐
A SPHERE!!!
๐=ยฑโ๐โ๐๐
CIRCLE!
Step Three: Determine what shape cross-sections (made perpendicular to x-axis) of the sphere are.
๐=โ๐โ๐๐
๐=โโ๐โ ๐๐
Step Five: SUM up the area of all the possible cross-sections. In calculus, we SUM using an INTEGRAL!!!
Step Six: Evaluate the integral.
Compare our answer above to the one we got using the geometry formula!!WE GET THE SAME ANSWER!
CALCULUS WORKS!!!
Step Four: Write the equation for the area of one of the cross-sections (in terms of x).
๐=โ๐โ๐๐
๐=โโ๐โ ๐๐
๐ด=๐ ๐2 ๐ด=๐ (โ 4โ๐ฅ2 )2 ๐ด=๐ (4โ๐ฅ2 )
Volume=
Volume=
=
=
=
โ33.51๐ข๐๐๐ก๐ 3
Example 2: Find the VOLUME of the solid formed by rotating the region bounded by the line around the x-axis on the interval [0,4].
What shape is this problem referring to?A CYLINDER!!!
Use GEOMETRY to find the volume of the cylinder.๐=๐ ๐2h
Use CALCULUS to find the volume of the cylinder.๐=โซ
0
4
๐ ๐2๐๐ฅ
๐=โซ0
4
๐ 22๐๐ฅ
๐=4๐ ๐ฅ|40
=
Example 3: Find the VOLUME of the solid formed by rotating the region bounded by the line around the x-axis on the interval
[0,1].
What shape is this problem referring to?
Use GEOMETRY to find the volume of the cylinder.
๐=13
๐๐2h
Use CALCULUS to find the volume of the cylinder.
๐=โซ0
1
๐ ๐2๐๐ฅ
๐=โซ0
1
๐ (2 ๐ฅ)2๐๐ฅ
๐=43
๐ ๐ข๐๐๐ก๐ 3
A CONE!!
๐=13
๐ (2 )2(1)
Consider what shape one cross-section (taken perpendicular to the x-axis) of the solid would be. A CIRCLE!!
๐=๐ ( 43 ๐ฅ3|10 )
So what happens the solid is not one we have a geometric formula
for?
2( )b
a
V f x dx โซ
You have to use CALCULUS!!! Hereโs the basic formula:
Radius of circular cross-
section
Example 4: Find the VOLUME of the solid formed by rotating the region bounded by the line around the x-axis on the interval
[-1,1].
2( )b
a
V f x dx โซ
๐=๐โซโ1
1
(๐ฅ3โ ๐ฅ+1 )2๐๐ฅ
Evaluate this using your calculatorโฆ
๐ โ6.76๐ข๐๐๐ก๐ 3
Example 6: Find the VOLUME of the solid formed by rotating the region bounded by the line around the x-axis on the interval .
๐=๐ โซ0
2๐
(2+sin ๐ฅ )2๐๐ฅ
๐=9๐ 2๐ข๐๐๐ก๐ 3
Example 5: Find the VOLUME of the solid formed by rotating the region bounded by the line , and around the y-axis .
ยฟ ๐โซโ 3
3
โ
Evaluate this using your calculatorโฆ
Because the circular cross-sections will be horizontal, we will integrate this time with respect to y! This means the bounds for integration should be y-values and the function must be solved for x.
๐=๐โซ๐
๐
( ๐ (๐ฆ ))2๐๐ฆ ( 16 ๐ฆ+ 12 )
2
๐๐ฆ
Example 6: Find the VOLUME of the solid formed by rotating the region bounded by the line , and around the y-axis .
โ74.16๐ข๐๐๐ก๐ 3
Example 7: Find the VOLUME of the solid formed by rotating the region bounded by the line , and the x-axis around the x-axis .
The key here is that you HAVE to use TWO integrals!
We need to determine EXACTLY where the functions intersect firstโฆโ๐ฅ=6โ๐ฅ๐ฅ=36โ12๐ฅ+๐ฅ2
0=๐ฅ2โ13๐ฅ+360=(๐ฅโ9)(๐ฅโ4)๐ฅ=4๐๐๐ 9
1. TO ROTATE OVER A LINE OTHER THAN ONE OF THE AXES.
2. TO ROTATE AN AREA NOT FORMED BY ONE OF THE AXES.
We still need to learnโฆ.
HOMEWORK
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