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Section B.6 Additional Formulas

Summation Formulas:.
n∑i=1c=cnn∑i=1i=n(n+1)2n∑i=1i2=n(n+1)(2n+1)6n∑i=1i3=(n(n+1)2)2
Trapezoidal Rule:.
∫baf(x)dx≈Δx2[f(x1)+2f(x2)+2f(x3)+⋯+2f(xn)+f(xn+1)]

with Error ≤(b−a)312n2[max|f″(x)|]

Simpson's Rule:.
∫baf(x)dx≈Δx3[f(x1)+4f(x2)+2f(x3)+4f(x4)+⋯+2f(xn−1)+4f(xn)+f(xn+1)]

with Error ≤(b−a)5180n4[max|f(4)(x)|]

Arc Length:.
L=∫ba√1+f′(x)2dx
Surface of Revolution:.
2π∫baf(x)√1+f′(x)2dx

(where f(x)≥0)

S=2π∫bax√1+f′(x)2dx

(where a,b≥0)

Work Done by a Variable Force:.
W=∫baF(x)dx
Force Exerted by a Fluid:.
F=∫bawd(y)ℓ(y)dy
Taylor Series Expansion for f(x):.
pn(x)=f(c)+f′(c)(x−c)+f″(c)2!(x−c)2+⋯+f(n)(c)n!(x−c)n+⋯
Maclaurin Series Expansion for f(x), where c=0:.
pn(x)=f(0)+f′(0)x+f″(0)2!x2+f‴(0)3!x3+⋯+f(n)(0)n!xn+⋯