Multiple Integrals

Geometrical Meaning z 35 30 25 20 15 10 5 5 -5 -10 6 4 2 -2 -4 -6 x y

Double Integrals over Rectangles If f (x,y) is a positive function, and R is a rectangle on the xy-plane, then the geometrical meaning of is the volume of the solid S that is above the rectangle R and is below the graph z = f (x,y).

Definition in terms of Riemann sum If f (x,y) is a function of two variables and R = [a,b]×[c,d] is a closed rectangle in the xy-plane, then we define if the limit exists, where the interval [a,b] is divided into m equal subintervals, the interval [c,d] is divided into n equal subintervals, A is the area of each sub-rectangle, (xij*,yij*) is any point in sub-rectangle ij.

Estimation of the integral If we know that m ≤ f (x,y) ≤ M on the rectangle R, then Midpoint Rule where xi is the midpoint of [xi-1, xi], and yj is the midpoint of [yj-1,yj]

Approximation of volume by Riemann Sum

y x equation of surface z = f(x,y) 2.0 1.5 0.5 0.5 0.4 0.6 0.2 0.7 0.8 1.0 0.5 0.5 0.4 0.6 0.2 0.7 0.8 -0.2 y 0.9 x -0.4 1.0

y x equation of surface z = f(x,y) 2.0 1.5 0.5 0.4 0.6 0.2 0.7 0.8 0.9 1.0 2.0 1.5 -0.4 -0.2 0.2 0.4 Δx

Iterated Integrals Definition The iterated integral means that we first integrate the function f with respect to y, treating x as a constant, and then integrate the result with respect to x. This can also be called iterated partial integration.

Fubini’s Theorem If f is continuous on the rectangle R = [a,b]×[c,d], then More generally, this is true if f is bounded on R, f is discontinuous only on a finite number of smooth curves, and the iterated integrals exist.

Properties of Double Integrals If D is the non-overlapping union of two regions D1 and D2, then If R = [a,b]×[c,d] is a rectangle and f (x,y) = g(x)h(y) is a product of two functions of one variable, then

Over General Regions y-simple (or type I) regions A plane region D is said to be y-simple if it lies between the graphs of two continuous functions of x. D = {(x,y): a ≤ x ≤ b, g1(x) ≤ y ≤ g2(x)} Example: x 1.5 1 0.5 -0.5 -1 -1.5 3 2.5 2

In this case x 1.5 1 0.5 -0.5 -1 -1.5 3 2.5 2

Example x 1.5 1 0.5 -0.5 -1 -1.5 3 2.5 2

x-simple (or type II) regions A plane region D is said to be x-simple if it lies between the graphs of two continuous functions of y. D = {(x,y): c ≤ x ≤ d, h1(y) ≤ x ≤ h2(y)} Example: x 5 4 3 2 1 -1 -2 -3 -4

In this case x 5 4 3 2 1 -1 -2 -3 -4

Example x 5 4 3 2 1 -1 -2 -3 -4

Non-simple Regions There are many regions that do not belong to any of the two previous types, such as the one indicated below. 10 8 6 4 2 12 In this case, we have to divide to region into several disjoint sub-regions so that each one is either x-simple or y-simple. The integral will then be the sum of several integrals, as stated in the next theorem. D1 D3 D2

Example: Evaluate 10 8 6 4 2 12

In Polar Coordinates r = b r = a Hence dA = r dr d dr r dθ  = β  = α Hence dA = r dr d

If f is continuous on a polar rectangle R given by 0 ≤ a ≤ r ≤ b, α ≤  ≤ β, where 0 ≤ β - α ≤ 2π then

If f is continuous on a polar region D of the form α ≤  ≤ β, where 0 ≤ β - α ≤ 2π 0 ≤ h1( ) ≤ r ≤ h2( ), then

Surface Area If S is a surface given by the continuously differentiable function f (x,y) over a region D, then its area is

Parametric Surface All surfaces we have seen so far can be described by equations of the form z = f (x,y), hence the surface can pass the vertical line test. In practice, many useful surfaces will not pass this test, and will not pass any horizontal line test either. In those cases, we cannot expect to describe the surface by functions of the form x = g(y,z) or y = h(x,z) etc. And the remedy is to introduce two new variables u and v that will control values of x, y, and z. The more precise definition is on the next page.

Parametric Surface Suppose that is a vector-valued function defined on a region D in the uv-plane. The set of all points (x,y,z) in 3 such that x = x(u,v), y = y(u,v), z = z(u,v), and (u,v) varies throughout D, is called a parametric surface.

Examples A sphere centered at the origin with radius 4. In principle this surface can be described by two equations but the result in plotting is not satisfactory due to vertical slopes near the edge of each hemisphere. (see next page)

A sphere centered at the origin with radius 4. It will be a lot better to use spherical coordinates

The curves that you can see on the sphere are the grid curves. The horizontal circles are created by keeping  fixed, for instance The vertical circles are created by keeping  fixed.

Example 2 A parallelogram with vertex P0 and two other corners A(a1, a2, a3) and B(b1, b2, b3). The parametric equations are

In many practical situations, we need to create the parametric equations for a given surface, and one useful method is to decide which directions the grid curves go first, and then determine what parameters to use. Finally create the equation for the surface (by adding vectors). This can be illustrated in the next example.

Equation for a Torus z b a φ θ x

The parametric equations for the previous Torus are x(,  ) = (b + a cos  )cos  y(,  ) = (b + a cos  )sin  z(,  ) = a sin  where 0 ≤  ,  ≤ 2π

Example 3 An ellipsoid The standard equation in rectangular coordinates is

y x One parameter will be angle  which starts from 0 to π. 1 Hence x = a cos 0.5 -0.5 -1 y -2 -1 1 1 -1 2 x

y x Another parameter will be angle  which starts from 0 to 2π. 1 For this ellipse, the y-max = b sin and z-max = c sin 0.5 -0.5 -1 y -2 -1 1 1 Hence y = (b sin ) cos and z = (c sin ) sin -1 2 x

Example 3 An ellipsoid In parametric form it can be (but not uniquely)

In some extreme cases, a surface will intersect itself In some extreme cases, a surface will intersect itself. If this happens, parametric equation is the only way to describe such as surface.

Example 5 A surface that intersects itself 4 2 -2 1 -4 2 0.2 3 0.4 0.6 0.8 1 1.2 1.4 4

Exercise: Match the following graphs with the equations on the next slide. II III y x z x z y y x z IV V VI 3

r(u,v) = cos v i + sin v j + uk r(u,v) = u cos v i + u sin v j + uk r(u,v) = u cos v i + u sin v j + vk r(u,v) = u3 i + u sin v j + u cos vk x = (u – sinu)cos v, y = (1 – cos u)sin v, z = u x = (1 – u)(3 + cosv)cos4πu, y = (1 – u)(3 + cosv)sin4πu, z = 3u + (1 – u)sin v

Surface area of Parametric Surfaces If a smooth parametric surface S is given by the equations and S is covered only once as (u,v) ranges throughout the domain D, then the surface area is