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shell method: revolved around y-axis
V=2π∫p(x)h(x)dx
shell method: revolved around x-axis
V=2π∫p(y)h(y)dy
arc length
L=∫√(1+(f’(x))²dx
surface area: revolved around x-axis
S=2π∫f(x)√(1+(f’(x))²dx
surface area: revolved around y-axis
S=2π∫x√(1+(f’(x))²dx
work
W=∫F(x)dx
Hooke’s Law
F=kx
Newton’s Law
F=C/x²
Boyle’s Law
P=k/V
mass
m=force/acceleration
moment
moment=mass∙distance
moment about the origin: one-dimensional system
M0=m1x1+…+mnxn
moment about the y-axis: two-dimensional system
My=m1x1+…+mnxn
moment about the x-axis: two-dimensional system
Mx=m1y1+…+mnyn
center of mass: two-dimensional system
(x̄=My/m,ȳ=My/m) where m=m1+…+mn
moment about the y-axis: planar lamina
My=∫x(f(x)-g(x))dx
moment about the x-axis: planar lamina
Mx=∫((f(x)+g(x))/2)(f(x)-g(x))dx
center of mass: planar lamina
(x̄=My/m,ȳ=My/m) where m=∫(f(x)-g(x))gx
pressure
P=density∙depth
fluid force on a horizontal surface
F=pressure∙area
force exerted by a force
F=∫h(y)L(y)dy