Lecture 10: Multiple Integrals 1. Key points Multiple integrals as Iterated Integrals Change of variables: Jacobian Maple int(f(x,y),[xa..b,yc..d]) VectorCalculus[Jacobian] LinearAlgebra[Determinant] 2. Multiple Integral as Iterated Integral General double integrals are expressed as where and specify a point in a plane and is the element of area in the plane. The integral is done over an area. Remark:. Type I: Constant integral limits Type I: Remark 1: We should avoid expressions like since we don't know which integral limits belong to which integral variables. (For this expresion, a common interpretation is the inner integral is with respect to and th eouter one is with.) Remark 2: The order of iterative integrals can be swapped as long as it does not change the integral value. (That is the case in most of physics applications. See the Fourier theoem for the case the order cannot be changed.) Example 1
b 2 0 0 A a Example 2 Type II: Variable integral limits Type II: Remark I: The order of integral cannot be swapped without the changing the integral limits.
Example 3 } with the integral area shown in Figure. a y There are many ways. We consider the following two methods. First integrate over by fixing and then integrate over. 0 0 a x First integrate over by fixing and then integrate over. Exercise 1 Evaluate the integral shown in Figure. for the area 4 2 0 0 3 6 Exercise 2 Find the area bounded by,, and. The crossing point of the red and blue lines is determined by. Its solution
is. The red line crosses the axis at and the blue line crosses the axis at. The area is bounded by the three lines. simplify 3 2 3. Change of variables Consider an integral over area with Cartesian coordinates and,. Now, we want to evaluate it with a new variable and. The relation between two sets of the variable is given by and. Then the element of area is given by where is Jacobian defined by Hence, the integral is evaluated as. Similarly, for three-dimensional integrals if the variables are transformed as,, and, the element of volume is given by where the Jacobian is defined as Example 4. Polar coordinates Integrate
over the circle of radius. Using the polar coordinates simplify r You can calculate Jacobian using VectorCalculus package. Exercise 3 Integrate over the region inside a sphere of radius. simplify Hence,. simplify 4. Examples in physics
Inertia tensor In order to investigate rotational dynamics of a solid object, we need to find the inertia tensor of the object. The inertia tensor is defined by where A common expression For example, a diagonal element is used. is given by and an off-diagonal element by where is the region the object occupies and the mass density at. Find the inertia tensor of a uniform solid cube of mass and side length with respect to the origin at a corner. The edges of the cube is parallel to the axes as shown in Figure.
Answer Since it is uniform, the density is constant:. 3 Taking into the cubic symmetry, all diagonal elements are identical and all off-diagonal elements are also identical. Hence, we obtain Center of mass The center of mass of an object is defined by Find the center of mass of a uniform solid cone of mass with height and base radius. Answer Using the cylindrical coordinates,, and. The corresponding Jacobian is given by (1) and its determinant is r. Therefore, Now, we evaluate the volume of the cone 3
Due to symmetry, the center of mass is on the axis. Thus, we need to calculate only. h 4 Hence the center of mass is radius of the base. from the base on the axis. Note that it does not depend on the Electric Field generated by electric charge Electric charge is distributed over the region with the charge density. Electric field at a position is given by If the net charge is uniformally distributed over a unit cube with a corner at the origin and the edges parallel to the axes, find the electric field at. (Point P in Figure)
Answer The charge density is Using the Cartesian coordinates, the component of the electric field is 1
simplify 1 Due to symmetry,. Homework: Due 10/9, 11am 10.1 Evaluate the integral over the triangle with vertices (0,0), (2,1), (3,0). 10.2 Evaluate the integral over the area bounded by,, and. 10.3 1. Find the center of mass of one quadrant of the disk with a radius. (See Figure) 10.4 A uniform solid disk with radius and thickness has mass. It is parallel to the plane and its center of mass coincides with the origin of the coordinates. Find its inertia tensor around the center of mass..