MA16200: Calculus II

Flash card of formulas for MA162: Calculus II

Sphere formula

(x-a)² + (y-b)² + (z-c)² = r²

Center and radius of sphere: (x-a)² + (y-b)² + (z-c)² = r²

center: (a, b, c) and radius: r

Equation of vector

a = (a₁, a₂, a₃)

Length of vector

|a| = √(a₁² + a₂² + a₃²)

Dot product of a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃)

a • b = (a₁ * b₁ + a₂ * b₂ + a₃ * b₃)

Dot product (in terms of angle and length)

a • b = |a||b|cosθ

Cross product of a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃)

a × b = (a₂ * b₃ - a₃ * b₂, a₃ * b₁ - a₁ * b₃, a₁ * b₂ - a₂ * b₁)

Length of cross product

|a × b| = |a||b|sinθ

project of m onto n (with angle)

projₙm = |m|cosθ * (n/ |n|)

project of m onto n (with dot product)

projₙm = (n • m)/|n| * (n/|n|)

area between two curves

A = ∫(ƒ(x) - g(x))dx

Volume by slicing

V = ∫A(x) dx

Volume by slicing (x-axis)

V = ∫πƒ(x)²dx

Volume by shells (y-axis)

V = ∫2πx * ƒ(x)dx

Arc length

L = ∫√(1 + ƒ'(x)²)dx

Surface area of rotating surface (x-axis)

S = ∫2πƒ(x) * √(1+ƒ'(x)²) dx

Work = force * distance

W = |F|cosθ * d

Hooke's law

F = k * x

Integration by part

∫udv = uv - ∫vdu

Useful sin and cos formulas

sin²x + cos²x = 1
cos²x = (1 + cos2x)/2
sin²x = (1-cos2x)/2

Useful tan and sec formulas

tan²x + 1 = sec²x
d/dx tanx = sec²x
d/dx secx = tanxsecx
d/dx tan⁻¹x = 1/(1+x²)

∫sinⁿx * cosⁿx dx

cosⁿx is odd: u = sinx
sinⁿx is odd: u = cosx
both are even: half angle formula

∫tanⁿx * secⁿx dx

secⁿx is even: u = tanx
tanx is odd: u = secx

Trigonometric substitution

√(a²-x²) -- x = asinθ
√(a²+x²) -- x = asecθ
√(x²-a²) -- x = atanθ

Improper integrals

∫∞ƒ(x)dx = lim∞∫ⁿƒ(x)dx