曲面面積為
∫∫dS = ∫∫√[1+(z'x)²+(z'y)²]dxdy,等號后面是二重積分,積分區(qū)域是所求曲面在平面xoy上的投影
z=√(x²+y²)與x+2z=3的交線在xoy上的投影是(3-x)² = 4(x²+y²)
即3(x+1)²+4y² = 12
解得y = √[3 - 3(x+1)²/4]
令x = 2cost - 1,則x的取值范圍是(-3,1),dx = -2sintdt,x從-3取到1時(shí),t從π取到0
∫∫dS = ∫∫√[1+(z'x)²+(z'y)²]dxdy
= ∫(-3,1)dx∫(-y(x),y(x))√[1+(z'x)²+(z'y)²]dy
= √2∫(-3,1)dx * 2y(x)
= 2√2∫(-3,1)dx * √[3 - 3(x+1)²/4]
= 2√2∫(π,0) (-2sint)dt * √3 sint
= -4√6∫(π,0) sin²tdt
= 2√6π