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AP Calculus AB. Day 7 Section 7.2. Theorem: The volume of a solid with cross-section of area A(x) that is perpendicular to the x-axis is given by. ***This circle is the base of a 3-D figure coming out of the screen. S. a. b. - PowerPoint PPT Presentation
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04/20/23 Perkins
AP Calculus AB
Day 7Section 7.2
Theorem: The volume of a solid with cross-section of area A(x) that is perpendicular to the x-axis is given by
b
a
V A x dx
Finding the volume of a solid with known cross-section is a 3-step process:
Step #1: Find the length (S) of the rectangle used to create the base of the figure.
Step #2: Find the area A(x) of each cross-section (in terms of this rectangle).
Step #3: Integrate the area function from the lower to the upper bound.
***This circle is the base of a 3-D figure coming out of the screen.
***This rectangle is a side of a geometric figure (a cross-section of the whole).
a b
Volume of a solid with cross-sections of area A(y) and perpendicular to the y-axis:
d
c
V A y dyc
d
S
S
Find the volume of the solid whose base is bounded by the circle with cross-sections perpendicular to the x-axis. These cross-sections are
a. squaresStep #1: Find S.
Step #2: Find A(x).
Step #3: Integrate the area function.
2 2 4x y
S
2 24 4S x x
2 24y x 24y x
22 4 x 2 2 4x y
-2 2
2squareA S
222 4A x x
24 4 x 216 4x S
2
2
2
16 4V x dx
2
2
0
2 16 4x dx 128
3 42.667
Find the volume of the solid whose base is bounded by the circle with cross-sections perpendicular to the x-axis. These cross-sections are
Step #1: Find S.
Step #2: Find A(x).
Step #3: Integrate.
b. equilateral triangles
2 2 4x y
S
22 4S x -2 2
2
equilateral triangle
3
4
SA
223
2 44
A x x
23 4 x
S
23
4S
2
2
2
3 4V x dx
2
2
0
2 3 4 x dx 32 3
3 18.475
Perkins
AP Calculus AB
Day 7Section 7.2
Theorem: The volume of a solid with cross-section of area A(x) that is perpendicular to the x-axis is given by
b
a
V A x dx
Finding the volume of a solid with known cross-section is a 3-step process:
a b
Find the volume of the solid whose base is bounded by the circle with cross-sections perpendicular to the x-axis. These cross-sections are
a. squaresStep #1: Find S.
Step #2: Find A(x).
Step #3: Integrate the area function.
2 2 4x y
Find the volume of the solid whose base is bounded by the circle with cross-sections perpendicular to the x-axis. These cross-sections are
Step #1: Find S.
Step #2: Find A(x).
Step #3: Integrate.
b. equilateral triangles
2 2 4x y
