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Midterm

MATH 2B Midterm: MATH2B, Chapter 6.1 term test 2Premium


Department
Mathematics
Course Code
MATH 2B
Professor
All
Study Guide
Midterm

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Section 6.2 Volumes
Volume
How would you take the 3D volume of a 2D function?
Rotate it around a line
Example
How would you take the volume of the area of = 2 1
2, 
around the x-axis?
What would it look like?
The volume created by the area rotated around the x-axis looks like a cone

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How can we find the volume?
You can use integrals to find the volume of the cone
What is the area of a circle?
2
What would the boundaries be?
The x and y intercepts are where the function intercepts with the other
lines
So = 0, = 4
What is the radius?
The function itself is the radius
= (2 1
2)
= 2
= 4
0(2 1
2)2
= 4
04 2 + 1
42
Here, you can move outside the integral and foil out (2 1
2)2
= (4 2 + 1
123) ( 0  4)
Then take the integral
= (16 16 +64
12) (0 0 + 0)
Plugging in the boundaries
= 16
3
Now you can check whether this is true using the actual formula for Volume of a cone
= 1
32
Plug in the necessary components
= 1
3()(2)2(4)

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= 16
3
Example # 1
Find the volume between the curves = and = 2 rotated around the 
What would it look like?
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