Rectification of plane curve when its equation is in polar form r=f(θ) 1. r = f(θ) $s = ∫↙{θ_1}↖{θ_2} √{r^2 + ({dr}/{dθ})^2} dθ$ 2. θ = f(r) $s = ∫↙{r_1}↖{r_2} √{1 + r^2 ({dθ}/{dr})^2} dθ$ 1. Cartesian curves $\table,\text"Equation of curve",\text"Formula in differential calculus", \text"Formula in integral calculus";1,y=f(x),ds = √{1 + ({dy}/{dx})^2} dx, s = ∫↙{x_1}↖{x_2} √{1 + ({dy}/{dx})^2} dx;2,x=g(y), ds = √{1 + ({dx}/{dy})^2} dy, s = ∫↙{y_1}↖{y_2} √{1 + ({dx}/{dy})^2} dy; 3,x = f_1 (t)\, y=f_2 (t), ds = √{({dx}/{dt})^2 + ({dy}/{dt})^2} dt, s = ∫↙{t_1}↖{t_2} √{({dx}/{dt})^2 + ({dy}/{dt})^2} dt$ 2. Polar Curves: $\table ,Equation of curve,Formula in ...
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Rule I: Integral with limits (a, b) as constants If I(α) = $∫↙{a}↖b f(x, α) dx, then {dI}/{dα} = ∫↙{a}↖b {δ}/{δα} f(x, α) dx$ Rule II: Integral with limits as functions of parameter: Leibnitz’s Rule ${dI}/{dα} = d/{dα}∫↙{a(α)}↖{b(α)} f(x, α) dx = ∫↙{a(α)}↖{b(α)} δ/{δα} f(x, α) dx + f(b, α) {db}/{dα} -f (a,α) {da}/{dα}$ Error function 1. $erf(x) = 2/{√π} ∫↙{0}↖x e^{-u^2} du$ 2. Complementary error function $erfc(x) = 2/{√π} ∫↙{x}↖∞ e^{-u^2} du $ 3. Alternate definition of error function $erf(x) = 1/{√π} ∫↙{0}↖{x^2} e^{-t} t^{- 1/2} dt$ Properties of error functions (1) erf(∞) = 1 (2) erf(0) = 0 ...
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Reduction formulae for sinusoidal functions 1. To find reduction formula for $∫ sin^n x dx$, n is positive integer ≥ 2 and evaluate completely $∫↙{0}↖{π/2} sin^n x dx$ $I_n = {n-1}/{n} I_{n-2}$ $\table ∫↙{0}↖{π/2} sin^n x dx,=,{n-1}/n {n-3}/{n-2} …. 3/4 1/2 π/2\; \text"if n is even";,=,{n-1}/n {n-3}/{n-2}…..4/5 2/3 1 \; \text"if n is odd"$ $\table ∫↙{0}↖{π/2} cos^n x dx,=,{n-1}/n {n-3}/{n-2} …. 3/4 1/2 π/2\; \text"if n is even";,=,{n-1}/n {n-3}/{n-2}…..4/5 2/3 1 \; \text"if n is odd"$ Additional Results: I. $∫↙{0}↖{π} sin^n x dx$ = $2 ∫↙{0}↖{π} sin^n x dx$ , for all n integral values of n II. $\table ∫↙{0}↖{π} ...
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Fourier Series f(x) = $a_o/2 + ∑↙{n =1 }↖∞ (a_n cos nx + b_n sin nx)$ $a_o, a_n, b_n$ are fourier coefficients Note (Useful Results) (i) $∫↙{c}↖{c + 2π} cosnx dx = [{sin nx}/{n}]^{c + 2π}_c$ = 0 (n ≠ 0) (ii) $∫↙{c}↖{c + 2π} sinnx dx = [-{cos nx}/{n}]^{c + 2π}_c$ = 0 (n ≠ 0) (iii) $\table∫↙{c}↖{c + 2π} sin mx cosnx dx,=, 1/2 ∫↙{c}↖{c + 2π} [sin(m + n)x + sin(m-n) x] dx;,=,1/2 [- {cos(m+n)x}/{(m+n)} – {cos (m-n)x}/{(m-n)}]_{c}^{c + 2π} = 0 (m≠n) $ If m=n, $∫↙{c}↖{c + 2π} sinmx cosnx dx = 1/2 ∫↙{c}↖{c + 2π} ...
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Rule to find the equation of orthogonal trajectories Note: Differentiate as many times as there are constants. For rectangular cartesian co-ordinates: Step 1: Given f(x,y,a) = 0, where a is a variable parameter. Step 2: Differentiate f(x,y,a) = 0 w.r.t. x and eliminate ‘a’. Step 3: Replace ${dy}/{dx}$ by $- {dx}/{dy}$ For polar co-ordinates: Step 1: Given f(r,θ,a) = 0, where a is a variable parameter. Step 2: Differentiate f(r,θ,${dr}/{dθ}$) = 0 w.r.t. x and eliminate ‘a’. Step 3: Replace ${dr}/{dθ}$ by ($- r^2{dθ}/{dr}$) Rate of decay of radioactive materials If u is amount of material at any time t, ...
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Order of differential equation Definition: The order of differential equation is the order of highest derivation that appears in the equation Degree of differential equation Definition: It is the degree of highest order differential coefficient or derivative, when differential coefficients are free from fractions General solution of differential equation A relation between the dependent and independent variables, which is free from derivatives, which satisfies a given differential equation which contains arbitrary constants equal to the order of the differential equation is called general solution or complete integral. Particular solution of a differential equation The solution obtained by assigning particular values ...
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List of Formulas (1) $∫ x^n dx = {x^{n+1}}/{n + 1} + c$ (2) $∫ 1/x dx = log|x| + c$ (3) $∫a^x dx = {a^x}/{loga} + c$ (4) $∫e^x dx = e^x + c$ (5) $∫sinx dx = – cosx + c$ (6) $∫cosx dx = sinx + c$ (7) $∫tanx dx = log |secx| + c$ (8) $∫cotx dx = log |sinx| + c$ (9) $∫secx dx = log |secx + tanx| + c$ (10) $∫cosecx dx = log |cosecx – cotx| + c$ (11) $∫sec^2 x dx = tanx + c$ (12) $∫cosec^2 x = – cotx ...
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Coefficient of Static Friction $μ_s = {F_s}/{R}$ …For impending motion Coefficient of Kinetic Friction $μ_k = {F_k}/{R}$ For actual motion R -> Magnitude of normal reaction Angle of Friction φ = $tan^{-1} μ $ For impending motion the angle between normal reaction and resultant reaction is called as angle of static friction. For actual motion when P = $F_k$ the angle between normal reaction and resultant reaction is called angle of kinetic friction. Angle of Respose Angle of inclined plane with horizontal at which body is just on verge of sliding (impending motion) and so $F_s$ = $μ_s$ R Angle ...
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There are as such no formulas in this chapter.
To find tension at extreme points of cable/at support: T = $R_A$ = $√{{A_x^2}+{A_y^2}}$ tan θ = ${A_y}/{A_x}$ For finding other tensions use Lami’s theorem at intermediate joints To find sag at any point Use geometry for example, ${A_y}/{A_x} = {\text"Verticle distance/sag"}/{\text"Horizontal distance"}$ Important tips 1. Observe number of unknown forces. If no. of unknown forces is ‘n’ then consider (n-3) parts of cable for finding the unknown quantities. 2. While selecting no. of parts care should be taken that parts must be starting from “same support” upto the points where sags of points are known. 3. Although the tensions ...
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