Mathematics Foundations/11.3 Multiple Integrals

11.3 Multiple Integrals Multiple integrals extend the concept of integration to functions of several variables, allowing us to calculate volumes, masses, and other quantities in multi-dimensional spaces. 11.3.1 Double Integrals A double integral represents the signed volume under a surface over a region in the xy-plane. Definition For a continuous function defined over a region , the double integral is defined as: where is the area of the -th subregion and is a point in that subregion. Evaluation Using Iterated Integrals For a type I region (defined by and ): For a type II region (defined by and ): Change of Order of Integration Sometimes, changing the order of integration can simplify the calculation: where the limits must be adjusted to describe the same region . 11.3.2 Double Integrals in Polar Coordinates For regions where polar coordinates are more convenient, we use the transformation: The area element becomes , giving: Applications 1. Area calculation: 2. Volume calculation: 3. Mass of a lamina: 4. Center of mass: , 11.3.3 Triple Integrals Triple integrals extend to functions of three variables over a three-dimensional region . Definition For a continuous function defined over a region in three-dimensional space: Evaluation in Rectangular Coordinates Other orders of integration are possible depending on the region's description. Triple Integrals in Cylindrical Coordinates Using the transformation: The volume element becomes : Triple Integrals in Spherical Coordinates Using the transformation: The volume element becomes : 11.3.4 Applications of Multiple Integrals Physical Applications 1. Volume: 2. Mass: 3. Center of mass: 4. Moments of inertia: Probability and Statistics For joint probability density functions or : 1. Probability over region : 2. Expected values: 3. Variance: 11.3.5 Problems 1. Evaluate where is the region bounded by . 2. Calculate the volume of the solid bounded by the paraboloid and the plane . 3. Find the mass of a solid hemisphere of radius with density function , where is a constant. 4. Evaluate where is the region bounded by the coordinate planes and the plane in the first octant. 5. Calculate the moment of inertia about the z-axis for a solid sphere of radius with constant density . 6. Use a triple integral to find the volume of the region bounded by the ellipsoid . 7. Find the center of mass of a hemispherical shell with constant density, inner radius , and outer radius . 8. Evaluate over the entire xy-plane and explain its significance in probability theory.