Expression 17: left bracket, "y" Subscript, 1 , Baseline less than "y" less than "y" Subscript, 2 , Baseline , right bracket Cannot use an inequality here.y1<y<y2
Method = 0 for the definite integral (volume under z = 0 is counted as negative)
Method = 1 for volume under the surface (volume under z = 0 is counted as positive)
"S" "u" "r" "f" "a" "c" "e" "A" "r" "e" "a"SurfaceArea
"S" Subscript, "u" "r" "f" "a" "c" "e" "A" "r" "e" "a" , BaselineSurfaceArea
r1c1:
r1c2: 68.3 9 8 6 1 468.398614
r1c3:
r2c1:
r2c2:
r2c3:
25
Formulas
Hide this folder from students.
26
Expression 27: "V" "o" "l" "u" "m" "e" equals Start integral from "y" Subscript, 1 , Baseline to "y" Subscript, 2 , Baseline , end integral, Start integral from "x" Subscript, 1 , Baseline to "x" Subscript, 2 , Baseline , end integral, "f" left parenthesis, "x" , "y" , right parenthesis "d" "x" "d" "y" Add slider for "o", "l", "u", and "m"Volume=∫y2y1∫x2x1fx,ydxdy
add slider:
27
Expression 28: "S" "u" "r" "f" "a" "c" "e" "A" "r" "e" "a" equals Start integral from "y" Subscript, 1 , Baseline to "y" Subscript, 2 , Baseline , end integral, Start integral from "x" Subscript, 1 , Baseline to "x" Subscript, 2 , Baseline , end integral, StartRoot, StartFraction, ∂ "f" Over ∂ "x" , EndFraction left parenthesis, "x" , "y" , right parenthesis squared positive StartFraction, ∂ "f" Over ∂ "y" , EndFraction left parenthesis, "x" , "y" , right parenthesis squared positive 1 , EndRoot "d" "x" "d" "y" Sorry, I don't understand the '∂' symbol.SurfaceArea=∫y2y1∫x2x1∂f∂xx,y2+∂f∂yx,y2+1dxdy
28
Perspective View
Hide this folder from students.
29
Expression 30: theta Subscript, "h" "o" "r" "i" "z" "o" "n" "t" "a" "l" , Baseline equals negative 0.5θhorizontal=−0.5
"S" "u" "r" "f" "a" "c" "e" "A" "r" "e" "a" equals Start integral from "y" Subscript, 1 , Baseline to "y" Subscript, 2 , Baseline , end integral, Start integral from "x" Subscript, 1 , Baseline to "x" Subscript, 2 , Baseline , end integral, StartRoot, StartFraction, ∂ "f" Over ∂ "x" , EndFraction left parenthesis, "x" , "y" , right parenthesis squared positive StartFraction, ∂ "f" Over ∂ "y" , EndFraction left parenthesis, "x" , "y" , right parenthesis squared positive 1 , EndRoot "d" "x" "d" "y" equals 68.4SurfaceArea=∫y2y1∫x2x1∂f∂xx,y2+∂f∂yx,y2+1dxdy=68.4
"V" "o" "l" "u" "m" "e" equals Start integral from "y" Subscript, 1 , Baseline to "y" Subscript, 2 , Baseline , end integral, Start integral from "x" Subscript, 1 , Baseline to "x" Subscript, 2 , Baseline , end integral, "f" left parenthesis, "x" , "y" , right parenthesis "d" "x" "d" "y" equals 137Volume=∫y2y1∫x2x1fx,ydxdy=137
