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Calc07_4day2

# Calc07_4day2 - y-axis 3 4 dx dy =-15 π = s S A rs = 3 5 =...

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7.4 Day 2 Surface Area Greg Kelly, Hanford High School, Richland, Washington (Photo not taken by Vickie Kelly)

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Surface Area: ds r Consider a curve rotated about the x -axis: The surface area of this band is: 2 r ds π⋅ The radius is the y -value of the function, so the whole area is given by: 2 b a y ds π This is the same ds that we had in the “length of curve” formula, so the formula becomes: Surface Area about x -axis (Cartesian): 2 2 1 b a dy S y dx dx = + To rotate about the y -axis, just reverse x and y in the formula!
4 4 3 y x = - + Example: Rotate about the y -axis. 4 4 3 y x - = - 3 3 4 y x - + = 3 3 4 x y = - + 3 4 dx dy = -

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Example: Rotate about the y -axis. 3 4 dx dy = - 2 4 0 3 3 2 3 1 4 4 S y dy π = - + + 4 0 3 25 2 3 4 16 y dy = - + 4 0 3 5 2 3 4 4 y dy = - + 4 2 0 5 3 3 2 8 y y = - + [ ] 5 6 12 2 = - + 5 6 2 = 15 = 3 3 4 x y = - +
4 4 3 y x = - + Example: Rotate about the

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Unformatted text preview: y-axis. 3 4 dx dy = -15 π = s . . S A rs = 3 5 = ⋅ ⋅ 15 = From geometry: → Example: y x = rotated about x-axis. 2 9 2 1 dy S y dx dx π = + ∫ 2 ( (1 ( , ) ^ 2), ,0,9) y d y x x + ∫ 117.319 ≈ → ENTER ( ) x ENTER Y STO Example: 2 2 1 y x = -2 1 y x =-12.5663706144 ≈ Check: rotated about x-axis. 2 2 1 x y + = 2 1 1 2 1 dy S y dx dx π- = + ∫ 2 ( (1 ( , ) ^ 2), , 1,1) y d y x x +-∫ 2 . . 4 S A r = 4 = 12.5663706144 ≈ → ENTER (1 ^ 2) x-ENTER Y STO Don’t forget to clear the x and y variables when you are done! π ENTER F4 4 , Y X Once again …...
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Calc07_4day2 - y-axis 3 4 dx dy =-15 π = s S A rs = 3 5 =...

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