Expression 16: StartFraction, "h" "n" Over 12 , EndFraction co tangent left parenthesis, StartFraction, pi Over "n" , EndFraction , right parenthesis left parenthesis, "S" squared positive "s" "S" positive "s" squared , right parenthesishn12cotπnS2+sS+s2
equals=
213.3 3 9 1 9 7 6 7 3213.339197673
16
Volume by Integration (Polygons)
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Expression 18: Start integral from 0 to "h" , end integral, StartFraction, "n" Over 4 , EndFraction left parenthesis, negative StartFraction, left parenthesis, "S" negative "s" , right parenthesis Over "h" , EndFraction "z" positive "S" , right parenthesis squared co tangent left parenthesis, StartFraction, pi Over "n" , EndFraction , right parenthesis "d" "z"∫h0n4−S−shz+S2cotπndz
equals=
213.3 3 9 1 9 7 6 7 3213.339197673
18
Volume by Integration (Shells)
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Expression 20: Start integral from 0 to StartFraction, "s" Over 2 , EndFraction cot left parenthesis, StartFraction, pi Over "n" , EndFraction , right parenthesis , end integral, 2tangent left parenthesis, StartFraction, pi Over "n" , EndFraction , right parenthesis "y" "h" times "n" "d" "y" positive Start integral from StartFraction, "s" Over 2 , EndFraction cot left parenthesis, StartFraction, pi Over "n" , EndFraction , right parenthesis to StartFraction, "S" Over 2 , EndFraction cot left parenthesis, StartFraction, pi Over "n" , EndFraction , right parenthesis , end integral, 2 "n" tangent left parenthesis, StartFraction, pi Over "n" , EndFraction , right parenthesis left parenthesis, negative StartFraction, "h" Over left parenthesis, StartNestedFraction, "S" NestedOver 2 , EndNestedFraction cot left parenthesis, StartNestedFraction, pi NestedOver "n" , EndNestedFraction , right parenthesis negative StartNestedFraction, "s" NestedOver 2 , EndNestedFraction cot left parenthesis, StartNestedFraction, pi NestedOver "n" , EndNestedFraction , right parenthesis , right parenthesis , EndFraction "y" positive left parenthesis, StartFraction, "h" Over left parenthesis, StartNestedFraction, "S" NestedOver 2 , EndNestedFraction cot left parenthesis, StartNestedFraction, pi NestedOver "n" , EndNestedFraction , right parenthesis negative StartNestedFraction, "s" NestedOver 2 , EndNestedFraction cot left parenthesis, StartNestedFraction, pi NestedOver "n" , EndNestedFraction , right parenthesis , right parenthesis , EndFraction times left parenthesis, StartFraction, "S" Over 2 , EndFraction cot left parenthesis, StartFraction, pi Over "n" , EndFraction , right parenthesis , right parenthesis , right parenthesis , right parenthesis "y" "d" "y"∫s2cotπn02tanπnyh·ndy+∫S2cotπns2cotπn2ntanπn−hS2cotπn−s2cotπny+hS2cotπn−s2cotπn·S2cotπnydy
equals=
213.3 3 9 1 9 7 6 7 3213.339197673
20
Surface Area
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21
Surface Area by Formula
22
Expression 23: StartFraction, "n" Over 4 , EndFraction left parenthesis, "S" squared cot left parenthesis, StartFraction, pi Over "n" , EndFraction , right parenthesis positive "s" squared cot left parenthesis, StartFraction, pi Over "n" , EndFraction , right parenthesis positive 2 StartRoot, left parenthesis, StartFraction, "S" Over 2 , EndFraction cot left parenthesis, StartFraction, pi Over "n" , EndFraction , right parenthesis , right parenthesis squared negative 2 left parenthesis, StartFraction, "s" Over 2 , EndFraction cot left parenthesis, StartFraction, pi Over "n" , EndFraction , right parenthesis , right parenthesis left parenthesis, StartFraction, "S" Over 2 , EndFraction cot left parenthesis, StartFraction, pi Over "n" , EndFraction , right parenthesis , right parenthesis positive left parenthesis, StartFraction, "s" Over 2 , EndFraction cot left parenthesis, StartFraction, pi Over "n" , EndFraction , right parenthesis , right parenthesis squared positive "h" squared , EndRoot left parenthesis, "S" positive "s" , right parenthesis , right parenthesisn4S2cotπn+s2cotπn+2S2cotπn2−2s2cotπnS2cotπn+s2cotπn2+h2S+s
equals=
235.6 1 9 1 0 1 0 0 9235.619101009
23
Surface Area by Integration (Polygons)
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Expression 25: Start integral from StartFraction, "s" Over 2 , EndFraction cot left parenthesis, StartFraction, pi Over "n" , EndFraction , right parenthesis to StartFraction, "S" Over 2 , EndFraction cot left parenthesis, StartFraction, pi Over "n" , EndFraction , right parenthesis , end integral, 2 "y" "n" tangent left parenthesis, StartFraction, pi Over "n" , EndFraction , right parenthesis StartRoot, 1 positive left parenthesis, negative StartFraction, "h" Over left parenthesis, StartNestedFraction, "S" NestedOver 2 , EndNestedFraction cot left parenthesis, StartNestedFraction, pi NestedOver "n" , EndNestedFraction , right parenthesis , right parenthesis negative left parenthesis, StartNestedFraction, "s" NestedOver 2 , EndNestedFraction cot left parenthesis, StartNestedFraction, pi NestedOver "n" , EndNestedFraction , right parenthesis , right parenthesis , EndFraction , right parenthesis squared , EndRoot "d" "y" positive Start integral from 0 to StartFraction, "s" Over 2 , EndFraction cot left parenthesis, StartFraction, pi Over "n" , EndFraction , right parenthesis , end integral, 2 "y" "n" tangent left parenthesis, StartFraction, pi Over "n" , EndFraction , right parenthesis "d" "y" positive Start integral from 0 to StartFraction, "S" Over 2 , EndFraction cot left parenthesis, StartFraction, pi Over "n" , EndFraction , right parenthesis , end integral, 2 "y" "n" tangent left parenthesis, StartFraction, pi Over "n" , EndFraction , right parenthesis "d" "y"∫S2cotπns2cotπn2yntanπn1+−hS2cotπn−s2cotπn2dy+∫s2cotπn02yntanπndy+∫S2cotπn02yntanπndy
equals=
235.6 1 9 1 0 1 0 0 9235.619101009
25
Perspective View
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26
Expression 27: theta Subscript, "h" "o" "r" "i" "z" "o" "n" "t" "a" "l" , Baseline equals negative 0.5θhorizontal=−0.5
