Department

MathematicsCourse Code

MATH 2BProfessor

AllStudy Guide

MidtermThis

**preview**shows pages 1-3. to view the full**12 pages of the document.**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

32

■ 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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