We can use Fubini’s theorem to write and evaluate a double integral as an iterated integral We can use a double Riemann sum to approximate the volume of a solid bounded above by a function of two variables over a rectangular region.

By taking the limit, this becomes a double integral representing the volume of the solid Properties of double integral are useful to simplify computation and find bounds on their values

Fubini’s theorem — if f ( x , y ) is a function of two variables that is continuous over a rectangular region R = { ( x , y ) ∈ ℝ 2 | a ≤ x ≤ b , c ≤ y ≤ d } , R = { ( x , y ) ∈ ℝ 2 | a ≤ x ≤ b , c…. Jacobian — the Jacobian J ( u , v ) in two variables is a 2 × 2 determinant: J ( u , v ) = | ∂ x ∂ u ∂ y ∂ u ∂ x ∂ v ∂ y ∂ v | ; J ( u , v ) = | ∂ x ∂ u ∂ y ∂ u ∂ x ∂ v ∂ y ∂ v | ; the…. one-to-one transformation — a transformation T : G → R defined as T ( u , v ) = ( x , y ) is said to be one-to-one if no two points map to the same image point. polar rectangle — the region enclosed between the circles r = a and r = b and the angles θ = α and θ = β ; it is described as R = { ( r , θ ) | a ≤ r ≤ b , α ≤ θ ≤ β } R = { ( r , θ ) | a ≤ r ≤ b….

Be able to use x = r cos θ , y = r sin θ , and d A = r d θ to convert an integral in rectangular coordinates to an integral in polar coordinates; use r 2 = x 2 + y 2 and θ = tan -1 ( y x ) to convert an integral in polar coordinates to an integral in rectangular coordinates, if needed.