Calc 2 Exam 2

x̄=

(total moment from x=0)/(total mass) OR M_y/m

ȳ=

(total moments from y=0)/(total mass) OR M_x/m

Center of mass=

(x̄,ȳ)

M_y equation

p∫x(f(x)-g(x))dx

M_y (conceptually)

M_y = mass * distance to y=axis from center of box (x)

M_x equation

(p/2) * ∫[(f(x))² - (g(x))²] dx

M_x conceptually

mass * distance to x-axis from center of box (1/2f(x)+g(x))

Midpoint Rule Equation

M_n = 𝚫x(f(x̄₁) + f(x̄₂) + … + f(x̄_n))

Midpoint rule conceptually

chose x̄_i to be the midpoint of the subinterval

Right hand rule equation

R_n = 𝚫x(fx₁) + f(x₂) + … + f(x_n))

Right hand rule conceptually

Chose x_i to be the right endpoint of the sub interval

Left hand rule equation

L_n = 𝚫x(f(x_o) + f(x₁) + … + f(x_n-1))

Left hand rule conceptually

Chose x_i to be the left endpoint of the subinterval

Trapezoidal rule equation

(𝚫x/2) * [(1)f(x_o) + (2)f(x₁) + (2)f(x₂) + … + (2)f(x_n-1) + (1)f(x_n)]

Which approximation is most accurate?

Midpoint, accounts for concavity

How to get a better approximation

More subintervals

When does L_n give a overestimate + underestimate?

Over when f(x) is dec.

Under when f(x) is inc.

When does R_n give a overestimate + underestimate?

Over when f(x) is inc.

Under when f(x) is dec.

When does T_n give a overestimate + underestimate?

Over if f(x) is concave up

Under if f(x) is concave down

E_t equation

E_t ≤ [k(b-a)³] / 12n^₂

Bounds of K

K ≤ lf(x)l for all x on a ≤ x ≤ b

Formula for mass

mass = ∫ (f(x)-g(x)) dx, same as area because mass = (area)(density)

Newtons law of cooling

T(t) = T_s + (T₀ - T_s)e^-kt

Eulers method

y_n+1 = y_n + h * f(x_n,y_n)

Population formula

P = P₀ * e^kt

how to find r

x² + y² = r²

how to find tan

y/x

how to find x

rcosΘ

how to find y

rsinΘ

linear form

dy/dx + P(x)y = Q(x)

I(x)

e^∫P(x)dx