※ 引述《smallprawn (水中瑕)》之銘言：
: 1. C is a smooth wire with coordinate functions x=sin(t),y=cos(t),
: and z=1 for 0≦t≦π/2. The density function along the wire is
: ψ(x,y)=xy(g/cm). Evaluate the mass of the wire.
四分之一圓
質量 = 密度乘上長度 (一維)
∫ xy ds
c
2 2
= ∫ sint cost √(cos(t) + sin(t) ) dt
c
π/2 1
= ∫ ── sin 2t dt
0 2
- 1 │π/2
= ── cos2t │
4 │0
( [-1] - [1]) 1
= - ─────── = ──
4 2
答案 0.5 g
: 2. Use polar coordinate to determine the volume of the solid that
: 2 2 2
: is under the hemispere x +y +z =1 and above the region R bounded
: 2 2
: by the circle x +y =y. (Note: Use the symmetry of the circle to
: the volume of the solid.)
2 2 2 2 2
x + y = y ， x + (y - 1/2) = ( 1/2 )
很普通的三重積分
1/2 1/2 + √(1/4 - x^2) √1-x^2-y^2
∫ ∫ ∫ dz dy dx
-1/2 1/2 - √(1/4 - x^2) 0
1/2 1/2 + √(1/4 - x^2)
∫ ∫ √1-x^2-y^2 dy dx
-1/2 1/2 - √(1/4 - x^2)
Polar Form
π sinθ
∫ ∫ √ 1 - r^2 r dr dθ
0 0
π -1 │sinθ
∫ ── (1 - r^2) │ dθ
0 3 │0
π -1 2
∫ ── (1 - sinθ - 1) dθ
0 3
π 1 2
∫ ── sin θ dθ
0 3
π 1 1 - cos2θ
∫ ── ────── dθ
0 3 2
π 1 1
∫ ── θ - ── sin2θ dθ
0 6 12
π
= ──
6
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◆ From: 140.118.234.83
※ 編輯: ntust661 來自: 140.118.234.83 (03/27 20:01)