六、积分学
1.常用的积分公式
∫ x k d x = 1 k + 1 x k + 1 + C , k ≠ − 1 \int x^kdx=\frac{1}{k+1} x^{k+1}+C,k\ne -1 ∫xkdx=k+11xk+1+C,k=−1
∫ 1 x d x = l n ∣ x ∣ + C \int \frac{1}{x} dx=ln\mid x\mid +C ∫x1dx=ln∣x∣+C
∫ e x d x = e x + C \int e^x dx=e^x +C ∫exdx=ex+C
∫ a x d x = a x l n a + C , a > 0 , a ≠ 1 \int a^x dx=\frac{a^x}{lna} +C,a>0,a\ne 1 ∫axdx=lnaax+C,a>0,a=1
∫ s i n x d x = − c o s x + C \int sinx dx=-cosx +C ∫sinxdx=−cosx+C
∫ c o s x d x = s i n x + C \int cosx dx=sinx +C ∫cosxdx=sinx+C
∫ t a n x d x = − l n ∣ c o s x ∣ + C \int tanx dx=-ln\mid cosx \mid +C ∫tanxdx=−ln∣cosx∣+C
∫ c o t x d x = l n ∣ s i n x ∣ + C \int cotx dx=ln\mid sinx \mid +C ∫cotxdx=ln∣sinx∣+C
∫ s e c x d x = ∫ d x c o s x = l n ∣ s e c x + t a n x ∣ + C \int secxdx=\int \frac{dx}{cosx}=ln\mid secx+tanx \mid +C ∫secxdx=∫cosxdx=ln∣secx+tanx∣+C
∫ c s c x d x = ∫ d x s i n x = l n ∣ c s c x − c o t x ∣ + C \int cscxdx=\int \frac{dx}{sinx}=ln\mid cscx-cotx \mid +C ∫cscxdx=∫sinxdx=ln∣cscx−cotx∣+C
∫ s e c 2 x d x = t a n x + C \int sec^2xdx=tanx +C ∫sec2xdx=tanx+C
∫ c s c 2 x d x = − c o t x + C \int csc^2xdx=-cotx +C ∫csc2xdx=−cotx+C
∫ s e c x t a n x d x = s e c x + C \int secx tanxdx=secx +C ∫secxtanxdx=secx+C
∫ c s c x c o t x d x = − c s c x + C \int cscx cotxdx=-cscx +C ∫cscxcotxdx=−cscx+C
∫ c s c x c o t x d x = − c s c x + C \int cscx cotxdx=-cscx +C ∫cscxcotxdx=−cscx+C
∫ 1 1 + x 2 d x = a r c t a n x + C \int \frac{1}{1+x^2} dx=arctanx +C ∫1+x21dx=arctanx+C
∫ 1 a 2 + x 2 d x = 1 a a r c t a n x a + C \int\frac{1}{a^2+x^2}dx=\frac{1}{a} arctan\frac{x}{a}+C ∫a2+x21dx=a1arctanax+C
∫ 1 1 − x 2 d x = a r c s i n x + C \int\frac{1}{\sqrt{1-x^2} }dx=arcsinx+C ∫1−x21dx=arcsinx+C
∫ 1 a 2 − x 2 d x = a r c s i n x a + C \int\frac{1}{\sqrt{a^2-x^2} }dx=arcsin\frac{x}{a} +C ∫a2−x21dx=arcsinax+C
∫ 1 x 2 + a 2 d x = l n ( x + x 2 + a 2 ) + C \int\frac{1}{\sqrt{x^2+a^2} }dx=ln(x+\sqrt{x^2+a^2} ) +C ∫x2+a21dx=ln(x+x2+a2)+C
∫ 1 x 2 − a 2 d x = l n ∣ x + x 2 − a 2 ∣ + C \int\frac{1}{\sqrt{x^2-a^2} }dx=ln\mid x+\sqrt{x^2-a^2}\mid +C ∫x2−a21dx=ln∣x+x2−a2∣+C
∫ 1 x 2 − a 2 d x = 1 2 a l n ∣ x − a x + a ∣ + C \int\frac{1}{x^2-a^2}dx=\frac{1}{2a}ln\mid \frac{x-a}{x+a}\mid+C ∫x2−a21dx=2a1ln∣x+ax−a∣+C
∫ 1 ( x + 1 ) 2 d x = x 2 ( 1 + x 2 ) − 1 2 a r c t a n x + C \int\frac{1}{(x+1)^2}dx=\frac{x}{2(1+x^2)}-\frac{1}{2}arctanx+C ∫(x+1)21dx=2(1+x2)x−21arctanx+C
∫ 1 a 2 − x 2 d x = 1 2 a l n ∣ x + a x − a ∣ + C \int\frac{1}{a^2-x^2}dx=\frac{1}{2a}ln\mid \frac{x+a}{x-a}\mid+C ∫a2−x21dx=2a1ln∣x−ax+a∣+C
∫ a 2 − x 2 d x = a 2 2 a r c s i n x a + x 2 a 2 − x 2 + C \int \sqrt{a^2-x^2} dx=\frac{a^2}{2}arcsin\frac{x}{a}+\frac{x}{2} \sqrt{a^2-x^2} +C ∫a2−x2dx=2a2arcsinax+2xa2−x2+C
∫ s i n 2 x d x = x 2 − s i n 2 x 4 + C \int sin^2x dx=\frac{x}{2}-\frac{sin2x}{4}+C ∫sin2xdx=2x−4sin2x+C
∫ c o s 2 x d x = x 2 + s i n 2 x 4 + C \int cos^2x dx=\frac{x}{2}+\frac{sin2x}{4}+C ∫cos2xdx=2x+4sin2x+C
∫ t a n 2 x d x = t a n x − x + C \int tan^2x dx=tanx-x+C ∫tan2xdx=tanx−x+C
∫ c o t 2 x d x = − c o t x − x + C \int cot^2x dx=-cotx-x+C ∫cot2xdx=−cotx−x+C
2.积分学推广公式\结论
(1)分步积分推广公式:
∫ u v ( n + 1 ) d x = u v ( n ) − u ′ v ( n − 1 ) + u ′ ′ v ( n − 2 ) − ⋯ + ( − 1 ) n u n v + ∫ u ( n + 1 ) v d x + C \int uv^{(n+1)} dx=uv^{(n)}-u'v^{(n-1)}+u''v^{(n-2)}-\dots+(-1)^{n}u^nv+\int u^{(n+1)}vdx+C ∫uv(n+1)dx=uv(n)−u′v(n−1)+u′′v(n−2)−⋯+(−1)nunv+∫u(n+1)vdx+C
(2)区间再现公式
∫ a b f ( x ) d x = ∫ a b f ( a + b − x ) d x \int_{a}^{b} f(x)dx=\int_{a}^{b} f(a+b-x)dx ∫abf(x)dx=∫abf(a+b−x)dx
(3)点火公式
∫ 0 π 2 s i n n x d x = ∫ 0 π 2 c o s n x d x = n − 1 n ⋅ n − 3 n − 2 ⋅ ⋯ ⋅ 2 3 ⋅ 1 ( n 为 大 于 1 的 奇 数 ) \int_{0}^{\frac{\pi}{2} }sin^nxdx=\int_{0}^{\frac{\pi}{2} }cos^nxdx=\frac{n-1}{n} \cdot \frac{n-3}{n-2} \cdot\dots \cdot\frac{2}{3} \cdot1(n为大于1的奇数) ∫02πsinnxdx=∫02πcosnxdx=nn−1⋅n−2n−3⋅⋯⋅32⋅1(n为大于1的奇数)
∫ 0 π 2 s i n n x d x = ∫ 0 π 2 c o s n x d x = n − 1 n ⋅ n − 3 n − 2 ⋅ ⋯ ⋅ 1 2 ⋅ π 2 ( n 为 正 偶 数 ) \int_{0}^{\frac{\pi}{2} }sin^nxdx=\int_{0}^{\frac{\pi}{2} }cos^nxdx=\frac{n-1}{n} \cdot \frac{n-3}{n-2} \cdot\dots \cdot\frac{1}{2} \cdot\frac{\pi }{2} (n为正偶数) ∫02πsinnxdx=∫02πcosnxdx=nn−1⋅n−2n−3⋅⋯⋅21⋅2π(n为正偶数)
∫ 0 π s i n n x d x = 2 ∫ 0 π 2 s i n n x d x ( n 为 正 整 数 ) \int_{0}^{\pi}sin^nxdx=2\int_{0}^{\frac{\pi}{2} }sin^nxdx(n为正整数) ∫0πsinnxdx=2∫02πsinnxdx(n为正整数)
∫ 0 π c o s n x d x = 0 ( n 为 正 奇 数 ) \int_{0}^{\pi}cos^nxdx=0(n为正奇数) ∫0πcosnxdx=0(n为正奇数)
∫ 0 π c o s n x d x = 2 ∫ 0 π 2 c o s n x d x ( n 为 正 偶 数 ) \int_{0}^{\pi}cos^nxdx=2\int_{0}^{\frac{\pi}{2} }cos^nxdx(n为正偶数) ∫0πcosnxdx=2∫02πcosnxdx(n为正偶数)
∫ 0 2 π s i n n x d x = ∫ 0 2 π c o s n x d x = 0 ( n 为 正 奇 数 ) \int_{0}^{2\pi}sin^nxdx=\int_{0}^{2\pi}cos^nxdx=0(n为正奇数) ∫02πsinnxdx=∫02πcosnxdx=0(n为正奇数)
∫ 0 2 π s i n n x d x = ∫ 0 2 π c o s n x d x = 4 ∫ 0 π s i n n x d x ( n 为 正 偶 数 ) \int_{0}^{2\pi}sin^nxdx=\int_{0}^{2\pi}cos^nxdx=4\int_{0}^{\pi}sin^nxdx(n为正偶数) ∫02πsinnxdx=∫02πcosnxdx=4∫0πsinnxdx(n为正偶数)
(4)积分与连续性
变限积分存在就必定连续.
(5)定积分定义式的推广
定义式: ∫ a b f ( x ) d x = lim n → ∞ ∑ i = 1 ∞ f ( a + b − a n i ) b − a n \int_{a}^{b} f(x)dx=\lim_{n \to \infty} \sum_{i=1}^{\infty} f(a+\frac{b-a}{n}i)\frac{b-a}{n} ∫abf(x)dx=limn→∞∑i=1∞f(a+nb−ai)nb−a
当a=0,b=1时: ∫ 0 1 f ( x ) d x = lim n → ∞ ∑ i = 1 ∞ f ( i n ) 1 n \int_{0}^{1} f(x)dx=\lim_{n \to \infty} \sum_{i=1}^{\infty} f(\frac{i}{n})\frac{1}{n} ∫01f(x)dx=limn→∞∑i=1∞f(ni)n1
2.常见的无法积分的不定积分
以下积分的结果不能用初等函数表示.
∫ s i n 1 x d x \int sin\frac{1}{x} dx ∫sinx1dx
∫ c o s 1 x d x \int cos\frac{1}{x} dx ∫cosx1dx
∫ s i n x x d x \int \frac{sinx}{x} dx ∫xsinxdx
∫ c o s x x d x \int \frac{cosx}{x} dx ∫xcosxdx
∫ t a n x x d x \int \frac{tanx}{x} dx ∫xtanxdx
∫ e x x d x \int \frac{e^x}{x} dx ∫xexdx
∫ s i n x 2 d x \int sinx^2dx ∫sinx2dx
∫ c o s x 2 d x \int cosx^2dx ∫cosx2dx
∫ t a n x 2 d x \int tanx^2dx ∫tanx2dx
∫ 1 l n x d x \int \frac{1}{lnx}dx ∫lnx1dx
∫ e a x 2 + b x + c d x \int e^{ax^2+bx+c}dx ∫eax2+bx+cdx
3.二重积分
∬ D f ( x , y ) d σ = ∫ a b d x ∫ φ 1 ( x ) φ 2 ( x ) f ( x , y ) d y \iint\limits_{D}f(x,y)d\sigma =\int_{a}^{b} dx\int_{\varphi_1(x)}^{\varphi_2(x)} f(x,y)dy D∬f(x,y)dσ=∫abdx∫φ1(x)φ2(x)f(x,y)dy
∬ D f ( x , y ) d σ = ∫ c d d y ∫ ψ 1 ( y ) ψ 2 ( y ) f ( x , y ) d y \iint\limits_{D}f(x,y)d\sigma =\int_{c}^{d} dy\int_{\psi _1(y)}^{\psi _2(y)} f(x,y)dy D∬f(x,y)dσ=∫cddy∫ψ1(y)ψ2(y)f(x,y)dy
∬ D f ( x , y ) d σ = ∫ α β d θ ∫ r 1 ( θ ) r 1 ( θ ) f ( r c o s θ , r s i n θ ) r d r \iint\limits_{D}f(x,y)d\sigma =\int_{\alpha }^{\beta } d\theta \int_{r_1(\theta )}^{r_1(\theta )} f(rcos\theta ,rsin\theta)\mathbf{ {\color{Red} r} } dr D∬f(x,y)dσ=∫αβdθ∫r1(θ)r1(θ)f(rcosθ,rsinθ)rdr
七、微分方程
1.变量可分离型
d y d x = f ( x ) g ( y ) ⇒ ∫ d y g ( y ) = f ( x ) d x \frac{dy}{dx} =f(x)g(y)\Rightarrow \int \frac{dy}{g(y)} =f(x)dx dxdy=f(x)g(y)⇒∫g(y)dy=f(x)dx
2.可化为变量分离型
d y d x = f ( a x + b y + c ) ⇒ 令 u = a x + b y + c , d u d x = a + b d y d x , 则 d u d x = a + b f ( u ) \frac{dy}{dx} =f(ax+by+c)\Rightarrow 令u=ax+by+c, \frac{du}{dx}=a+b\frac{dy}{dx},则\frac{du}{dx} =a+bf(u) dxdy=f(ax+by+c)⇒令u=ax+by+c,dxdu=a+bdxdy,则dxdu=a+bf(u)
d y d x = φ ( y x ) ⇒ 令 u = y x , d y d x = u + x d u d x , 则 d u φ ( u ) − u = d x x \frac{dy}{dx} =\varphi(\frac{y}{x})\Rightarrow 令u=\frac{y}{x}, \frac{dy}{dx} =u+x\frac{du}{dx} ,则\frac{du}{\varphi (u)-u} =\frac{dx}{x} dxdy=φ(xy)⇒令u=xy,dxdy=u+xdxdu,则φ(u)−udu=xdx
3.一阶线性微分方程
形如 y ′ + p ( x ) y = q ( x ) y'+p(x)y=q(x) y′+p(x)y=q(x)的方程,通解为:
y = e − ∫ p ( x ) d x [ ∫ e ∫ p ( x ) d x ⋅ q ( x ) d x + C ] y=e^{-\int p(x)dx}[\int e^{\int p(x)dx}\cdot q(x)dx+C] y=e−∫p(x)dx[∫e∫p(x)dx⋅q(x)dx+C]
4.伯努利方程
形如 y ′ + p ( x ) y = q ( x ) y n y'+p(x)y=q(x)y^n y′+p(x)y=q(x)yn的方程,令 z = y 1 − n z=y^{1-n} z=y1−n, d z d x = ( 1 − n ) y − n d y d x \frac{dz}{dx} =(1-n)y^{-n}\frac{dy}{dx} dxdz=(1−n)y−ndxdy,原式化为:
1 1 − n ⋅ d z d x + p ( x ) z = q ( x ) \frac{1}{1-n} \cdot \frac{dz}{dx} +p(x)z=q(x) 1−n1⋅dxdz+p(x)z=q(x)
5. y ′ ′ = f ( x , y ′ ) y''=f(x,y') y′′=f(x,y′)(不显含y)型方程
形如 y ′ ′ = f ( x , y ′ ) y''=f(x,y') y′′=f(x,y′)的方程,令 y ′ = p ( x ) y'=p(x) y′=p(x),原式化为 d p d x = f ( x , p ) \frac{dp}{dx}=f(x,p) dxdp=f(x,p),积分得 p = φ ( x , C 1 ) p=\varphi (x,C_1) p=φ(x,C1),通解为:
y = ∫ φ ( x , C 1 ) d x + C 2 y=\int\varphi (x,C_1)dx+C_2 y=∫φ(x,C1)dx+C2
6. y ′ ′ = f ( y , y ′ ) y''=f(y,y') y′′=f(y,y′)(不显含x)型方程
形如 y ′ ′ = f ( y , y ′ ) y''=f(y,y') y′′=f(y,y′)的方程,令 y ′ = p ( x ) y'=p(x) y′=p(x),则 y ′ ′ = d p d x = d p d y ⋅ p y''=\frac{dp}{dx}=\frac{dp}{dy}\cdot p y′′=dxdp=dydp⋅p,原式化为 p d p d y = f ( y , p ) p\frac{dp}{dy}=f(y,p) pdydp=f(y,p);解得 p = φ ( y , C 1 ) p=\varphi (y,C_1) p=φ(y,C1),即 d y d x = φ ( y , C 1 ) \frac{dy}{dx}=\varphi (y,C_1) dxdy=φ(y,C1),化简积分得:
∫ d y φ ( y , C 1 ) = x + C 2 \int\frac{dy}{\varphi (y,C_1)}=x+C_2 ∫φ(y,C1)dy=x+C2
7.二阶常系数齐次线性微分方程
形如 y ′ ′ + p y ′ + q y = 0 y''+py'+qy=0 y′′+py′+qy=0的方程,特征方程为 r 2 + p r + q = 0 r^2+pr+q=0 r2+pr+q=0,则
特征根 | 通解 |
---|---|
p 2 − 4 q > 0 p^2-4q>0 p2−4q>0,两个不等实根 | y = C 1 e r 1 x + C 2 e r 2 x y=C_1e^{r_1x}+C_2e^{r_2x} y=C1er1x+C2er2x |
p 2 − 4 q = 0 p^2-4q=0 p2−4q=0,两个相等实根 | y = ( C 1 + C 2 x ) e r x y=(C_1+C_2x)e^{rx} y=(C1+C2x)erx |
p 2 − 4 q < 0 p^2-4q<0 p2−4q<0,一对共轭复根 | y = e α x ( C 1 c o s β x + C 2 s i n β x ) y=e^{\alpha x}(C_1cos\beta x+C_2sin\beta x) y=eαx(C1cosβx+C2sinβx) |
8.二阶常系数非齐次线性微分方程
形如 y ′ ′ + p y ′ + q y = f ( x ) y''+py'+qy=f(x) y′′+py′+qy=f(x)的方程,通解=齐次微分方程通解+非齐次微分方程的一个特解。其中特解形式为:
(1)当自由项 f ( x ) = P n ( x ) e α x f(x)=P_n(x)e^{\alpha x} f(x)=Pn(x)eαx的形式时,特解设为: y ∗ = e α x Q n ( x ) x k y^*=e^{\alpha x}Q_n(x)x^k y∗=eαxQn(x)xk.
其中:
- α \alpha α与自由项的 α \alpha α相同;
- Q n ( x ) Q_n(x) Qn(x)是与 P n ( x ) P_n(x) Pn(x)同阶的一般n次多项式,例如 P n ( x ) = x 2 + x P_n(x)=x^2+x Pn(x)=x2+x应设 Q n ( x ) = a x 2 + b x + c Q_n(x)=ax^2+bx+c Qn(x)=ax2+bx+c;
- k k k根据特征根与 α \alpha α的关系而定,若 α \alpha α不是特征根,k=0;若 α \alpha α是单特征根,k=1;若 α \alpha α是二重特征根,k=2;
(2)当自由项 f ( x ) = e α x [ P m ( x ) c o s β x + P n ( x ) s i n β x ] f(x)=e^{\alpha x}[P_m(x)cos\beta x+P_n(x)sin\beta x] f(x)=eαx[Pm(x)cosβx+Pn(x)sinβx]时,特解设为:
y ∗ = e α x [ Q l ( 1 ) ( x ) c o s β x + Q l ( 2 ) ( x ) s i n β x ] x k y^*=e^{\alpha x}[Q^{(1)}_l(x)cos\beta x+Q^{(2)}_l(x)sin\beta x]x^k y∗=eαx[Ql(1)(x)cosβx+Ql(2)(x)sinβx]xk
其中:
- α , β \alpha,\beta α,β与自由项的 α , β \alpha,\beta α,β相同;
- Q l ( x ) Q_l(x) Ql(x)是 ( l ∈ m a x [ m , n ] ) (l\in max[m,n]) (l∈max[m,n])的 l l l阶一般多项式, l l l就是取 P ( x ) P(x) P(x)和 Q ( x ) Q(x) Q(x)的最高阶。
- k k k根据特征根与 α \alpha α的关系而定,若 α ± β i \alpha\pm \beta i α±βi不是特征根,k=0;若 α ± β i \alpha\pm \beta i α±βi是特征根,k=1;
9.欧拉方程
形如 x 2 d 2 y d x 2 + p x d y d x + q y = f ( x ) x^{2} \frac{d^2y}{dx^2}+px\frac{dy}{dx} +qy=f(x) x2dx2d2y+pxdxdy+qy=f(x)的方程,
(1)当 x > 0 x>0 x>0时,令 x = e t x=e^t x=et,则 t = l n x t=lnx t=lnx, d t d x = 1 x \frac{dt}{dx} =\frac{1}{x} dxdt=x1
则 d y d x = d y d t ⋅ d t d x = 1 x ⋅ d y d t \frac{dy}{dx} =\frac{dy}{dt} \cdot \frac{dt}{dx} =\frac{1}{x} \cdot \frac{dy}{dt} dxdy=dtdy⋅dxdt=x1⋅dtdy
d 2 y d x 2 = − 1 x 2 ⋅ d y d t + 1 x 2 ⋅ d 2 y d t 2 \frac{d^2y}{dx^2} =-\frac{1}{x^2} \cdot \frac{dy}{dt}+\frac{1}{x^2} \cdot \frac{d^2y}{dt^2} dx2d2y=−x21⋅dtdy+x21⋅dt2d2y
原式化为:
d 2 y d x 2 + ( p − 1 ) d y d t + q y = f ( e t ) \frac{d^2y}{dx^2} +(p-1)\frac{dy}{dt}+qy=f(e^t) dx2d2y+(p−1)dtdy+qy=f(et)
八、微积分的应用
1.定积分计算平面图形面积
直角坐标系下: S = ∫ a b ∣ y 1 ( x ) − y 2 ( x ) ∣ d x S=\int_{a}^{b} \mid y_1(x)-y_2(x)\mid dx S=∫ab∣y1(x)−y2(x)∣dx
极坐标系下: S = 1 2 ∫ α β ∣ r 1 2 ( θ ) − r 2 2 ( θ ) ∣ d θ S=\frac{1}{2} \int_{\alpha }^{\beta } \mid r_1^2(\theta )-r_2^2(\theta)\mid d\theta S=21∫αβ∣r12(θ)−r22(θ)∣dθ
参数方程换元: S = ∫ a b y ( x ) d x = ∫ α β y ( t ) x ′ ( t ) d t S=\int_{a}^{b}y(x) dx= \int_{\alpha }^{\beta }y(t)x'(t)dt S=∫aby(x)dx=∫αβy(t)x′(t)dt
2.定积分计算旋转体体积
(1)绕x轴旋转
V = π ∫ a b y 2 ( x ) d x V= \pi \int_{a }^{b } y^2(x)dx V=π∫aby2(x)dx
V = π ∫ a b ∣ y 1 2 ( x ) − y 2 2 ( x ) ∣ d x V=\pi \int_{a}^{b} \mid y_1^2(x)-y_2^2(x)\mid dx V=π∫ab∣y12(x)−y22(x)∣dx
(2)绕y轴旋转
V y = 2 π ∫ a b x ∣ y ( x ) ∣ d x V_y=2\pi \int_{a}^{b} x\mid y(x)\mid dx Vy=2π∫abx∣y(x)∣dx
V y = 2 π ∫ a b x ∣ y 1 ( x ) − y 2 ( x ) ∣ d x V_y=2\pi \int_{a}^{b} x\mid y_1(x)-y_2(x)\mid dx Vy=2π∫abx∣y1(x)−y2(x)∣dx
3.相关变化率
若 y = f ( x ) , x = x ( t ) , y = y ( t ) y=f(x),x=x(t),y=y(t) y=f(x),x=x(t),y=y(t)均可导,则 d y d x = d y d x ⋅ d x d t = f ′ ( x ) d x d t \frac{dy}{dx}=\frac{dy}{dx} \cdot \frac{dx}{dt} =f'(x)\frac{dx}{dt} dxdy=dxdy⋅dtdx=f′(x)dtdx
4.曲率
曲率公式:
k = ∣ y ′ ′ ∣ [ 1 + ( y ′ ) 2 ] 3 2 k=\frac{\mid y''\mid }{[1+(y')^2]^{\frac{3}{2} } } k=[1+(y′)2]23∣y′′∣
曲率半径:
R = 1 k = [ 1 + ( y ′ ) 2 ] 3 2 ∣ y ′ ′ ∣ R=\frac{1}{k} =\frac{[1+(y')^2]^{\frac{3}{2} }}{\mid y''\mid } R=k1=∣y′′∣[1+(y′)2]23
参数方程 x = φ ( t ) , y = ψ ( t ) x=\varphi (t),y=\psi (t) x=φ(t),y=ψ(t)曲率公式:
k = ∣ φ ′ ( t ) ψ ′ ′ ( t ) − φ ′ ′ ( t ) ψ ′ ( t ) ∣ [ φ ′ 2 ( t ) + ψ ′ 2 ( t ) ] 3 2 k=\frac{\mid \varphi '(t)\psi ''(t)-\varphi ''(t)\psi' (t)\mid }{[\varphi '^2(t)+\psi'^2 (t)]^{\frac{3}{2} } } k=[φ′2(t)+ψ′2(t)]23∣φ′(t)ψ′′(t)−φ′′(t)ψ′(t)∣
5.积分求功
(1)变力沿直线做功
W = ∫ a b F ( x ) d x W=\int_{a}^{b} F(x)dx W=∫abF(x)dx
(2)抽水做功
W = ρ g ∫ a b x A ( x ) d x W=\rho g\int_{a}^{b} xA(x)dx W=ρg∫abxA(x)dx
6.形心
x ˉ = ∬ D x d σ ∬ D d σ = ∫ a b x f ( x ) d x ∫ a b f ( x ) d x \bar{x} =\frac{\iint\limits_{D}xd\sigma }{\iint\limits_{D}d\sigma}=\frac{\int_{a}^{b}xf(x)dx }{\int_{a}^{b}f(x)dx} xˉ=D∬dσD∬xdσ=∫abf(x)dx∫abxf(x)dx
y ˉ = ∬ D y d σ ∬ D d σ = 1 2 ∫ a b f 2 ( x ) d x ∫ a b f ( x ) d x \bar{y} =\frac{\iint\limits_{D}yd\sigma }{\iint\limits_{D}d\sigma}=\frac{\frac{1}{2} \int_{a}^{b}f^2(x)dx }{\int_{a}^{b}f(x)dx} yˉ=D∬dσD∬ydσ=∫abf(x)dx21∫abf2(x)dx
7.弧长
d s = ( d x ) 2 + ( d y ) 2 ds=\sqrt{(dx)^2+(dy)^2} ds=(dx)2+(dy)2
s = ∫ a b 1 + [ y ′ ( x ) ] 2 d x s=\int_{a}^{b} \sqrt{1+[y'(x)]^2} dx s=∫ab1+[y′(x)]2dx
s = ∫ a b 1 + [ y ′ ( x ) ] 2 d x s=\int_{a}^{b} \sqrt{1+[y'(x)]^2} dx s=∫ab1+[y′(x)]2dx
s = ∫ α β [ x ′ ( t ) ] 2 + [ y ′ ( t ) ] 2 d t s=\int_{\alpha }^{\beta } \sqrt{[x'(t)]^2+[y'(t)]^2} dt s=∫αβ[x′(t)]2+[y′(t)]2dt
s = ∫ α β [ r ′ ( θ ) ] 2 + [ r ′ ( θ ) ] 2 d θ s=\int_{\alpha }^{\beta } \sqrt{[r'(\theta )]^2+[r'(\theta )]^2} d\theta s=∫αβ[r′(θ)]2+[r′(θ)]2dθ
8.旋转体表面积
s = 2 π ∫ a b ∣ y ( x ) ∣ 1 + [ y ′ ( x ) ] 2 d x s=2\pi \int_{a}^{b} \mid y(x) \mid \sqrt{1+[y'(x)]^2} dx s=2π∫ab∣y(x)∣1+[y′(x)]2dx
s = 2 π ∫ α β ∣ y ( t ) ∣ [ x ′ ( t ) ] 2 + [ y ′ ( t ) ] 2 d t s=2\pi \int_{\alpha }^{\beta } \mid y(t) \mid \sqrt{[x'(t)]^2+[y'(t)]^2} dt s=2π∫αβ∣y(t)∣[x′(t)]2+[y′(t)]2dt
9.傅里叶级数
(1)定义
以 2 l 2l 2l为周期的函数 f ( x ) f(x) f(x),满足一定条件,其傅里叶级数处处收敛,其和函数为 S ( x ) S(x) S(x),有:
S ( x ) = a 0 2 + ∑ n = 1 ∞ ( a n c o s n π l x + b n s i n n π l x ) S(x)=\frac{a_0}{2} +\sum_{n=1}^{\infty } (a_ncos\frac{n\pi }{l}x+b_nsin\frac{n\pi}{l}x) S(x)=2a0+n=1∑∞(ancoslnπx+bnsinlnπx)
a n = 1 l ∫ l − l f ( x ) c o s n π l x d x , ( n = 0 , 1 , 2 ⋯ ) a_n=\frac{1}{l} \int_{l}^{-l} f(x)cos\frac{n\pi }{l} xdx,(n=0,1,2\cdots) an=l1∫l−lf(x)coslnπxdx,(n=0,1,2⋯)
b n = 1 l ∫ l − l f ( x ) s i n n π l x d x , ( n = 1 , 2 ⋯ ) b_n=\frac{1}{l} \int_{l}^{-l} f(x)sin\frac{n\pi }{l} xdx,(n=1,2\cdots) bn=l1∫l−lf(x)sinlnπxdx,(n=1,2⋯)
S ( x ) = S(x)= S(x)= | - |
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f ( x ) f(x) f(x) | x为连续点 |
f ( x − 0 ) + f ( x + 0 ) 2 \frac{f(x-0)+f(x+0)}{2} 2f(x−0)+f(x+0) | x为间断点 |
f ( − l + 0 ) + f ( l − 0 ) 2 \frac{f(-l+0)+f(l-0)}{2} 2f(−l+0)+f(l−0) | x = ± l x=\pm l x=±l |
(2)奇偶性
当f(x)在 [ − l , l ] [-l,l] [−l,l]上是奇函数时, a n = 0 a_n=0 an=0,展开式只含正弦函数,称之为正弦级数.
当f(x)在 [ − l , l ] [-l,l] [−l,l]上是偶函数时, b n = 0 b_n=0 bn=0,展开式只含余弦函数,称之为余弦级数.