Calc 2 Midterm 2

disk method

V = ∫[a to b] πf(x)² dx

washer method

V = π∫[a to b] R²-r² dx

shells method

V = ∫[a to b] 2πrf(x) dx

average value on [a, b]

1/(b-a) ∫[a to b] f(x) dx

work

∫[a to b] F(x) dx (m = pV)

average x-value x̅ (M(y)/m)

(∫[a to b] xf(x) dx) / (∫[a to b] f(x) dx)

average y-value ȳ (M(x)/m)

½ (∫[a to b] f(x)²) / ∫[a to b] f(x) dx

area under a parametric curve

A = ∫[α to β] y(t)x′(t) dt

arc length

∫[a to b] √((dx/dt)² + (dy/dt)²) dt

convert from polar to cartesian

x = rcos(θ), y = rsin(θ)

convert from cartesian to polar

θ = arctan(y/x), r = √(x² + y²)

area under a polar curve

½ ∫[θ1 to θ2] f(θ)² dθ

In a sequence, if lim(n → ∞) |aₙ| = 0, then…

lim(n → ∞) aₙ = 0

In a sequence, if lim(n → ∞) aₙ = L and f is continuous, then…

lim(n → ∞) f(aₙ) = f(lim(n → ∞) aₙ) = f(L)

A sequence {aₙ} is increasing if…

aₙ₊₁ > aₙ

A sequence {aₙ} is decreasing if…

aₙ₊₁ < aₙ

Monotonic Convergence Theorem

Bounded monotonic sequences converge.

slope of a line tangent to a parametric curve

dy/dx