math 113 midterm 2

cartesian arc length

∫a,b sqrt(1+f’(x))² dx

cartesian surface of rev x-axis

∫a,b 2pi y sqrt(1+f’(x))² dx

cartesian surface of rev y-axis

∫a,b 2pi x sqrt(1+f’(x))² dx

cartesian surface of rev y = c

∫a,b 2pi |y-c| sqrt(1+f’(x))² dx

cartesian surface of rev x = d

∫a,b 2pi |x-d| sqrt(1+f’(x))² dx

Fluid pressure

∫a,b weight * Function for depth at y * function for length at y dy

Centroid x

1/A ∫a,b x (f(x)-g(x)) dx

Centroid y

1/A ∫a,b ½ (f(x)² - g(x)²) dx

Parametric slope

dy/dx = (dy/dt) / (dx/dt)

Second derivative parametric

d²y/dx² = d/dt*(dy/dt) / dx/dt

horizontal asymptote

dy/dt = 0

vertical asymptote

dx/dt = 0

Parametric arclength

∫t0,t1 sqrt (f’(t)² - g’(t)²) dx

parametric rotated surface area about x-axis

∫t0,t1 2pi g(t) sqrt (f’(t)² - g’(t)²) dx

parametric rotated surface area about y-axis

∫t0,t1 2pi f(t) sqrt (f’(t)² - g’(t)²) dx

parametric area with x-axis

∫t0,t1 f’(t)g(t)

parametric area with y-axis

∫t0,t1 f(t)g’(t)

polar cords x= y= r = theta =

x=r*cos(theta)

y=r*sin(theta)

r = sqrt(x²+y²)

theta = arctan(y/x) when x>0

theta = arctan(y/x) + pi when x<0

polar cords arclength

∫theta0,theta1 sqrt (r + (dr/d(theta))²) dx

area of polar cords

∫theta0,theta1 ½ r²

∫theta0,theta1 ½ (r1² - r2²)

Slope of polar cords

dy/dx = (dr/d(theta)sin - rcos) / (dr/d(theta)cos+rsin)