Section 16.7 Surface Integrals

Surface Integrals We now consider integrating functions over a surface S that lies above some plane region D.

Surface Integrals Let’s suppose the surface S is described by z=g(x,y) and we are considering a function f(x,y,z) defined on S The surface integral is given by: Where dS is the change in the surface AREA!

More useful: i.e. we convert a surface integral into a standard double integral that we can compute!

Example Evaluate Where And S is the portion of the plane 2x+y+2z=6 in the first octant.

Surface Integrals We still need to discuss surface integrals of vector fields…but we need a few new notions about surfaces first…. Recall the vector form of a line integral (which used the tangent vector to the curve): For surface integrals we will make use of the normal vector to the surface!

Normal vector to a surface If a surface S is given by z=g(x,y), what is the normal vector to the surface at a point (x,y,g(x,y)) on the surface?

Definition: Oriented Surface Suppose our surface has a tangent plane defined at every point (x,y,z) on the surface Then at each tangent plane there are TWO unit normal vectors with n 1 = -n 2 If it is possible to choose a unit normal vector n at every point (x,y,z) so that n varies continuously over S, we say S is an oriented surface

Example Positive orientation Negative orientation Remark: An oriented surface has two distinct sides

Surface Integrals of Vector Fields If F is a continuous vector field defined on an oriented surface S with unit normal vector n, then the surface integral of F over S is This is often called the flux of F across S

Using our knowledge of the normal vector and the surface area, this can be simplified…

i.e. a more simplified look at this…

An application If is the density of a fluid that is moving through a surface S with velocity given by a vector field, F(x,y,z), then Represents the mass of the fluid flowing across the surface S per unit of time.

Example Let S be the portion of the paraboloid Lying above the xy-plane oriented by an upward normal vector. A fluid with a constant density is flowing through the surface S according to the velocity field F(x,y,z) =. Find the rate of mass flow through S.