Overview
DOUBLE/TRIPLE INTEGRALS IN CARTESIAN COORDINATES
1. Consider a quadrilateral region R in the x-y plane with vertices (1,3), (2,1), (3,6) and (5,4). Use double-integration to calculate the area of this region. Hint: you will have to choose the order of stacking: along y first or along x first (or decide on the “slices” you pick in R). Either way, the region R will have to be split up in three parts R1, R2 and R3, each with their own boundaries. 2. Continuing with the same region R as in question 1. Imagine now a cylinder in the positive z- direction built on this region. The top of the cylinder however is intersected by a plane given by the equation z = 5x – 3y + 15. Calculate the volume of the resulting cylinder bounded by this plane and by the x-y plane (“the floor”). 3. Continuing with the cylinder from question 2. If the material of this cylinder was the same throughout the volume, that is, if the mass density were constant, then the mass of the cylinder would simply be the volume times the mass density. Note: Mass density, typically symbolized by the Greek letter ?, is the amount of mass per unit volume of a material. Since an infinitesimal cube has volume dV = dxdydz, it follows that the mass dm in this cube is given by dm = ?dxdydz. The total mass of an object is obviously the sum (= integral) of the masses dm of all its parts: ? = ????? = ????(?,?,?)?? = ????(?,?,?)?????? (the boundaries of the triple integrals depend on how the object’s volume is described) This integral can be solved by a triple stacking process of the little cubes dxdydz. Depending on how complicated the shape of the object is and on how complicated the mass density varies inside the object, the order of stacking (x first?, then y? finally z? or ....?) can simplify the
calculation or make it harder – this is a matter of trial and error. Remember that with 3 variables there are 6 different orders that can be considered. a) I want you to now calculate the mass M of the cylinder from question 2, knowing that the mass density only depends on x and y as follows: ?(?,?) = 1 + 2? + ?^2 . Note that this could be done by means of a triple stacking process, but also by means of a double-stacking process closely related to the integral from question 2. b) I want you to now calculate the mass M of the cylinder from question 2, knowing that the mass density is given by ?(?) = 1 + ?. c) I want you to now calculate the mass M of the cylinder from question 2, knowing that the mass density is given by ?(?) = 1 + ? + ?^2 +? DOUBLE INTEGRAL IN POLAR COORDINATES 4. Consider the following region R in the x-y plane: the region outside the circle with radius 2 centered on the origin (0,0) and inside the circle given by the equation (in polar coordinates) ? = 4???? (which is a circle with radius 2 centered on (2,0)). Calculate the average temperature across this region, if you are given that the temperature varies across this region according to the following equation: ?(?,?) = 2??.