Calculus.ppt

7/27/2019 Calculus.ppt

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Calculus Notes•

Volumes by slicing can be found by adding up eachslice of the solid as the thickness of the slices gets

smaller and smaller, in other words

the number of slices goes to infinity.

•To find the volume of each slice of the pyramid youwould find the area of each square then multiply thearea by the thickness of the slice. The thickness would

be dx because we are slicing with respect to the x-axis. Next you would add the volumes of each slice together to find the total volume.

•This would be a Riemann sum with the limit as n thenumber of slices going to infinity.



• Only now instead of f(x*) we will be summing the

areas of the base of each slice. So change

f(x*)in the Riemann Sum to A(x*)whichrepresents the area of the base.

n

k

b

a

k k x

dx x f x x f A1

*

0max)()(lim

n

k

b

ak k

x dx x A x x AV 1

*

0max )()(lim



Here is an example using a

cylinder

Find the volume of the cylinder

using the formula and slicing

with respect to the x -axis.

A = r 2

A = 22

= 4

6

2

4 dx

16824)2(4)6(44 62 x



• Now use the formula you learned in

Geometry to find the area

• V= r 2 h=(22)(4) = 16

You can also work problems likethese with respect to the y -axis.



• Volume of solids of revolution using

disks

• If you take the area under the line y = x

from 0 to 4 it will look like the diagram

below

4

If you rotate this area around the x-axis

it will form a cone (see below diagram)

4



• Now use the formula below to find the

volume of the 3-D figure formed byrotating around the x-axis.

• http://www.plu.edu/~heathdj/java/calc2/Solid.html

• This method is called disks when revolved

around the x-axis ( note: it is sliced with

respect to the x-axis and is revolved

around the x-axis)

http://www.plu.edu/~heathdj/java/calc2/Solid.html

http://www.plu.edu/~heathdj/java/calc2/Solid.html



• Formula for finding the volume of the solid

formed when f(x) is revolved around the x-

axis

• Each slice is a circle, the formula for the

area of a circle is A= r 2. The radius is the

y-value of the function AKA f(x). So, the

area formula becomes A= (f(x))2

• Substitute into the formula

dx x f dx x A

b

a

b

a 2)()(



• Equation y = x from 0 to 4

With geometry V = 1/3 (42)(4)=64/3

4

• With Calculus

3

64

03

64

3)(

4

0

34

0

2

x

dx x



Your turnp. 410 # 6. find the volume of the solid that

results when the region enclosed by the

given curves is revolved around the x-axis

y = sec x, x = , x = , y = 04

3

Use the formula dx x f

b

a

2

)(

4tan3tantansec3

4

3

4

2

xdx x

313

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Calculus.ppt