Question

球体的体积计算公式微积分推导

Answer 1

圆：x²+y²=r²,(注意,r为常数）

x² = (r² - y²) ——— [1]

切片面积:A = π x² ——— [2]

切片体积:

用[2]的结果

δv = A * δy

δv = π x² δy,用[1]的结果

δv = π (r² - y²) δy

v = ∫{[π (r² - y²)],-r,r} dy

v = π ∫{[(r² - y²)],-r,r} (提出常数)

v = 2π∫{[ (r² - y²)],0,r} (-r到0 和 0到r 对称)

v = 2π [y*r² - y³/3]

v = 2π {[r * r² - r³/3]-0} (极限代入y)

v = 2π {[r³ - r³/3]-0}

v = 4/3 π r³