Ch 14 Formulas

volume over rectangles

∫∫ f(x, y) dx dy

area of rectangle

∫∫ dy dx

average value of f over R

1/area of R * ∫∫ f dA

volume of circle

∫∫ f(r, θ) r dr dθ

area of a circle

∫∫ r dr dθ

volume in cylindrical coordinates

∫∫∫ f(r, θ, z) r dz dr dθ

volume in spherical coordinates

∫∫∫ p² sin∅ dp d∅ dθ

cylindrical to rectangular: x =

r cosθ

cylindrical to rectangular: y =

r sinθ

spherical to rectangular: x =

p sin∅ cosθ

spherical to rectangular: y =

p sin∅ sinθ

spherical to rectangular: z =

p cos∅

spherical to cylindrical: r =

p sin∅

spherical to cylindrical: z =

p cos∅

∫ cos²x dx

½ + ¼ sin(2x) + c

∫ sin²x dx

½ - ¼ sin(2x) + c

∫ tan²x dx

tan(x) - x + c

∫ csc²x dx

-cot(x) + c

∫ sec²x dx

tan(x) + c

∫ cot²x dx

-cot(x) - x + c