平面图形的面积
直角坐标系
- y = f ( x ) y=f(x) y=f(x) 与 x x x轴, x 1 = a x_1=a x1=a, x 2 = b x_2=b x2=b 围成的区域面积 S = ∫ a b ∣ f ( x ) ∣ d x S=\int_a^b |f(x)|dx S=∫ab∣f(x)∣dx
- x = ϕ ( y ) x=\phi(y) x=ϕ(y) 与 y y y轴, y 1 = c y_1=c y1=c, y 2 = d y_2=d y2=d 围成的区域面积 S = ∫ c d ∣ ϕ ( y ) ∣ d y S=\int_c^d |\phi(y)|dy S=∫cd∣ϕ(y)∣dy
- y = f ( x ) y=f(x) y=f(x) 与 y = g ( x ) y=g(x) y=g(x), x 1 = a x_1=a x1=a, x 2 = b x_2=b x2=b 围成的区域面积 S = ∫ a b ∣ f ( x ) − g ( x ) ∣ d x S=\int_a^b |f(x)-g(x)|dx S=∫ab∣f(x)−g(x)∣dx
- x = ϕ ( y ) x=\phi(y) x=ϕ(y) 与 x = ψ ( y ) x=\psi(y) x=ψ(y), y 1 = c y_1=c y1=c, y 2 = d y_2=d y2=d 围成的区域面积 S = ∫ c d ∣ ϕ ( y ) − ψ ( y ) ∣ d y S=\int_c^d |\phi(y)-\psi(y)|dy S=∫cd∣ϕ(y)−ψ(y)∣dy
参数方程
- y = ψ ( t ) y=\psi(t) y=ψ(t), x = ϕ ( t ) x=\phi(t) x=ϕ(t), 与参数 t = α t=\alpha t=α, t = β t=\beta t=β 围城的区域面积\
- S = ∫ α β ∣ ψ ( t ) d ϕ ( t ) ∣ = ∫ α β ∣ ψ ( t ) ϕ ′ ( t ) ∣ d t S=\int_\alpha^\beta |\psi(t)d \phi(t)|=\int_\alpha^\beta |\psi(t) \phi'(t)|d t S=∫αβ∣ψ(t)dϕ(t)∣=∫αβ∣ψ(t)ϕ′(t)∣dt
极坐标系
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r=r(θ), 与夹角
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θ=β 围城的区域面积
S = 1 2 ∫ α β r 2 ( θ ) d θ S=\frac{1}{2}\int_\alpha^\beta r^2(\theta) d\theta S=21∫αβr2(θ)dθ
几何体的体积
平行截面的函数
- 平行于 x O z xOz xOz 截面的面积 A ( y ) A(y) A(y), 体积为 V = ∫ a b A ( y ) d y V=\int_a^b A(y)dy V=∫abA(y)dy
- 平行于 y O z yOz yOz 截面的面积 A ( x ) A(x) A(x), 体积为 V = ∫ a b A ( x ) d x V=\int_a^b A(x)dx V=∫abA(x)dx
旋转体
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y = f ( x ) y=f(x) y=f(x) 与 x x x轴, x = a x=a x=a, x = b x=b x=b 的图形围绕 x x x 轴旋转， 旋转体的体积为 V = π ∫ a b f ( x ) 2 d x V=\pi\int_a^b f(x)^2dx V=π∫abf(x)2dx
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y = f ( x ) y=f(x) y=f(x) 与 y = g ( x ) y=g(x) y=g(x)轴, x = a x=a x=a, x = b x=b x=b 的图形围绕 x x x 轴旋转， 旋转体的体积为 V = π ∫ a b f ( x ) 2 − g ( x ) 2 d x V=\pi\int_a^b f(x)^2-g(x)^2dx V=π∫abf(x)2−g(x)2dx ( g ( x ) ≤ f ( x ) , ∀ x ∈ [ a , b ] g(x)\leq f(x), \forall x\in[a,b] g(x)≤f(x),∀x∈[a,b])
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x = f ( y ) x=f(y) x=f(y) 与 y y y轴, y = a y=a y=a, y = b y=b y=b 的图形围绕 y y y 轴旋转， 旋转体的体积为 V = π ∫ a b f ( y ) 2 d y V=\pi\int_a^b f(y)^2dy V=π∫abf(y)2dy
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x = f ( y ) x=f(y) x=f(y) 与 x = g ( y ) x=g(y) x=g(y)轴, y = a y=a y=a, y = b y=b y=b 的图形围绕 y y y 轴旋转， 旋转体的体积为 V = π ∫ a b f ( y ) 2 − g ( y ) 2 d y V=\pi\int_a^b f(y)^2-g(y)^2dy V=π∫abf(y)2−g(y)2dy ( g ( y ) ≤ f ( y ) , ∀ y ∈ [ a , b ] g(y)\leq f(y), \forall y\in[a,b] g(y)≤f(y),∀y∈[a,b])
曲线的弧长
- 直角坐标系 y = f ( x ) y=f(x) y=f(x), s = ∫ a b 1 + ( y ′ ) 2 d x s=\int_a^b \sqrt{1+(y')^2}dx s=∫ab1+(y′)2 dx
- 参数方程 x = ϕ ( t ) , y = ψ ( t ) x=\phi(t),y=\psi(t) x=ϕ(t),y=ψ(t), s = ∫ a b ( ϕ ′ ) 2 + ( ψ ′ ) 2 d t s=\int_a^b \sqrt{(\phi')^2+(\psi')^2}dt s=∫ab(ϕ′)2+(ψ′)2 dt
- 极坐标 r = r ( θ ) r=r(\theta) r=r(θ), s = ∫ α β r 2 + ( r ′ ) 2 d θ s=\int_\alpha^\beta \sqrt{r^2+(r')^2} d\theta s=∫αβr2+(r′)2 dθ
旋转体的侧面积
- y = f ( x ) y=f(x) y=f(x) 围绕 x x x 轴旋转, S = 2 π ∫ a b ∣ f ( x ) ∣ 1 + ( f ′ ( x ) ) 2 d x S=2\pi \int_a^b |f(x)|\sqrt{1+(f'(x))^2}dx S=2π∫ab∣f(x)∣1+(f′(x))2 dx