FORMULA SHEET FOR MAT187
Trigonometric Identities.
• cos2 (x) =
1 + cos(2x)
2
• sin2 (x) =
1 − cos(2x)
2
• sin(x) = cos
π
2
−x
Applications of integration.
Z bq
2
1 + f 0 (x) dx
• Arc length for y = f (x)
a
b
Z
• Area of a surface of revolution y = f (x) revolved around x−axis
q
2
2πf (x) 1 + f 0 (x) dx
a

x
−x

 sinh(x) = e − e
2 −x
x

 cosh(x) = e + e
2
• Hyperbolic Functions
Numerical Integration.
Z
• Midpoint Rule
b
f (x) dx ≈
a
n
X
xi−1 + xi
∆x
f
2
i=1
2
b−a
∆x max f 00 (x)
24
x∈[a,b]
Z b
∆x
f (x) dx ≈ f (a) + 2f (x1 ) + 2f (x2 ) + · · · + 2f (xn−1 ) + f (b)
2
a
|EM | 6
• Trapezoid Rule
2
b−a
∆x max f 00 (x)
12
x∈[a,b]
Z b
∆x
f (x) dx ≈ f (a) + 4f (x1 ) + 2f (x2 ) + · · · + 4f (xn−1 ) + f (b)
3
a
|ET | 6
• Simpson’s Rule (n even)
|ES | 6
4
b−a
∆x max f 0000 (x)
180
x∈[a,b]