Department

MathematicsCourse Code

MATH 2BProfessor

AllStudy Guide

MidtermThis

**preview**shows pages 1-2. to view the full**7 pages of the document.**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 x,x xis and y xisy = − 2

1 − a−a

around the x-axis?

●What would it look like?

○The volume created by the area rotated around the x-axis looks like a cone

●How can we find the volume?

Volumes

Section 6.2

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○You can use integrals to find the volume of the cone

●What is the area of a circle?

○rπ2

○What would the boundaries be?

■The x and y intercepts are where the function intercepts with the other

lines

■So 0, x 4x= =

○What is the radius?

■The function itself is the radius

■ (2 x)r= − 2

1

● πrΔxV = 2

○ (2 ) dxV = ∫

4

0

π − 2

12

○π 2x x dx= ∫

4

0

4 − + 4

1 2

○Here, you can move outside the integral and foil out π 2 )( − 2

12

○π(4x x x) (between 0 and 4)= − 2+ 1

12 3

■Then take the integral

○π(16 6 ) π(0 0 0)= − 1 + 12

64 − − +

■Plugging in the boundaries

○=3

16π

●Now you can check whether this is true using the actual formula for Volume of a cone

○ πr HV = 3

12

■Plug in the necessary components

○ (π)(2) (4)V= 3

12

○ V=3

16π

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