Find points of intersection:
4 + (y − 7)² = 13
(y − 7)² = 9
y − 7 = ± 3
y = 7 ± 3
y = 4 or 10
Points of intersection: (13,4) and (13,10)
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Use cylindrical shell method: V = 2π ∫ [a to b] (r * h) dy
where: a = 4, b = 10, r = y, h = 13 − (4+(y−7)²)
V = 2π ∫ [4 to 10] y * (13−(4+(y−7)²)) dy
V = 2π ∫ [4 to 10] (−y³+14y²−40y) dy
V = 504π
Or use washer method: V = π ∫ [a to b] (R²−r²) dx
where a = 4 (x = 4 at vertex of parabola), b = 13
R = 7 + √(x−4) .... top of parabola
r = 7 − √(x−4) .... bottom of parabola
V = π ∫ [4 to 13] (7+√(x−4))² − (7−√(x−4))² dx
V = π ∫ [4 to 13] 28√(x−4) dx
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Answers & Comments
Find points of intersection:
4 + (y − 7)² = 13
(y − 7)² = 9
y − 7 = ± 3
y = 7 ± 3
y = 4 or 10
Points of intersection: (13,4) and (13,10)
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Use cylindrical shell method: V = 2π ∫ [a to b] (r * h) dy
where: a = 4, b = 10, r = y, h = 13 − (4+(y−7)²)
V = 2π ∫ [4 to 10] y * (13−(4+(y−7)²)) dy
V = 2π ∫ [4 to 10] (−y³+14y²−40y) dy
V = 504π
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Or use washer method: V = π ∫ [a to b] (R²−r²) dx
where a = 4 (x = 4 at vertex of parabola), b = 13
R = 7 + √(x−4) .... top of parabola
r = 7 − √(x−4) .... bottom of parabola
V = π ∫ [4 to 13] (7+√(x−4))² − (7−√(x−4))² dx
V = π ∫ [4 to 13] 28√(x−4) dx
V = 504π