二重积分

Double Integral

曲顶柱体的体积，平面薄片的质量，基本思路：二重积分转为二次积分

\iint_{D} f (x, y) d σ = lim_{λ \to 0} \sum_{i = 1}^{n} f (ξ_{i}, η_{i}) Δ σ_{i}

\begin{array}{r} \iint_{D} f (x, y) d σ = \iint_{D_{1}} f (x, y) d σ + \iint_{D_{2}} f (x, y) d σ \end{array}

\begin{array}{r} \iint_{D} f (x, y) d σ = f (ξ, η) σ \end{array}

\begin{array}{r} | \iint_{D} f (x, y) d σ | \leq \iint_{D} | f (x, y) | d σ \end{array}

积分区域 $D$ 关于 $y$ 轴对称， $D_{1}$ 为积分的右半部分

\begin{array}{r} f (- x, y) = - f (x, y) \Rightarrow \iint_{D} f (x, y) = 0 \end{array}

\begin{array}{r} f (- x, y) = f (x, y) \Rightarrow \iint_{D} f (x, y) = 2 \iint_{D_{1}} f (x, y) \end{array}

积分区域 $D$ 关于 $x$ 轴对称， $D_{1}$ 为积分的上半部分

\begin{array}{r} f (x, - y) = - f (x, y) \Rightarrow \iint_{D} f (x, y) = 0 \end{array}

\begin{array}{r} f (x, - y) = f (x, y) \Rightarrow \iint_{D} f (x, y) = 2 \iint_{D_{1}} f (x, y) \end{array}

积分区域 $D$ 关于原点对称

\begin{array}{r} f (- x, - y) = - f (x, y) \Rightarrow \iint_{D} f (x, y) = 0 \end{array}

\begin{array}{r} f (- x, - y) = f (x, y) \Rightarrow \iint_{D} f (x, y) = 2 \iint_{D_{1}} f (x, y) \end{array}

确定好积分的区域，想明白积分的次序。以单次积分的理解将二重积分转为二次积分，最后转为定积分的计算

\begin{array}{r} \iint_{D} f (x, y) d σ = \int_{a}^{b} A (x) d x = \int_{a}^{b} [\int_{φ_{1} (x)}^{φ_{2} (x)} f (x, y) d y] d x \end{array}

\begin{array}{r} \iint_{D} f (x, y) d σ = \int_{a}^{b} A (y) d y = \int_{c}^{d} [\int_{ψ_{1} (y)}^{ψ_{2} (y)} f (x, y) d x] d y \end{array}

\begin{array}{r} {\begin{cases} x = ρ \cos θ \\ y = ρ \sin θ \end{cases} \Rightarrow d x d y \to ρ d ρ d θ \end{array}

\begin{aligned} \iint_{D} f (ρ \cos θ, ρ \sin θ) ρ d ρ d θ & = \int_{α}^{β} d θ \int_{φ_{1} (θ)}^{φ_{2} (θ)} f (ρ \cos θ, ρ \sin θ) ρ d ρ \end{aligned}