Lec 25: Triple integrals in rectangular & cylindrical | MIT 18.02 Multivariable Calculus, Fall 2007

This paragraph introduces the transition from studying double integrals and line integrals in a plane to triple integrals in space. The lecturer emphasizes the similarity in concepts but notes the additional complexity due to the extra coordinate. It encourages students to review material from previous weeks to ensure a solid understanding before moving on to space integrals. The paragraph concludes with an introduction to triple integrals, explaining how they are used to calculate the integral over a function of three variables, f(x, y, z), over a region in space, using volume elements dV. The importance of choosing the right order of integration is hinted at, and the paragraph sets the stage for upcoming examples.

The lecturer delves into the specifics of setting up triple integrals, using the example of a region between two paraboloids. The paragraph explains the importance of identifying the region of integration and the function to be integrated. It discusses the choice of integrating first over z due to the ease of establishing bounds for z given x and y. The paragraph also touches on the symmetry of the problem and the potential for simplifying calculations using cylindrical coordinates, which will be discussed later. The focus is on understanding the setup of triple integrals rather than the complexity of the function being integrated.

This section continues the exploration of the region between two paraboloids, focusing on visualizing the solid and understanding its geometric properties. The lecturer describes the top and bottom surfaces of the solid, which are the paraboloids themselves, and the sides, which form a circular cross-section due to the symmetry of the rotation about the z-axis. The paragraph also discusses the process of setting up the triple integral by choosing an order of integration and emphasizes the importance of understanding the bounds for each variable. The goal is to find the volume of the region, and the paragraph concludes with the setup for a triple integral in rectangular coordinates.

The paragraph discusses the process of projecting the solid onto the xy-plane to determine the bounds for the variables x and y in the triple integral. It explains the concept of the shadow of the solid in the xy-plane and how it forms a disk. The lecturer then describes how to find the radius of this disk by determining the intersection of the two paraboloids. The paragraph also explains how to set up the middle integral for y, given a fixed value of x, and the outer integral for x. The focus is on understanding the bounds for the integral and the process of projection to simplify the setup.

The lecturer introduces the transition from rectangular coordinates to cylindrical coordinates to simplify the triple integral. The paragraph explains the concept of cylindrical coordinates, where r represents the distance from the z-axis and theta represents the angle from the x-axis. It discusses the advantages of using cylindrical coordinates due to the symmetry of the problem and how it simplifies the volume element in the integral. The paragraph also touches on the idea of using polar coordinates for the projection on the xy-plane while keeping z as it is, setting the stage for a more straightforward evaluation of the integral.

This paragraph provides a deeper understanding of cylindrical coordinates, explaining how they are used to locate a point in space. It describes the conversion between rectangular (x, y, z) and cylindrical coordinates (r, theta, z) and the implications of these coordinates for defining surfaces in space, such as cylinders. The paragraph also discusses the implications of the convention that r is always positive and how it affects the definition of vertical planes in cylindrical coordinates. It concludes with a brief mention of the potential to use symmetry in calculations.

The paragraph discusses the various calculations that can be performed using triple integrals, such as finding the volume, mass, average value of a function, center of mass, and moment of inertia of a solid. It explains the concept of density and how it relates to mass, as well as the formulas for calculating the average value of a function and the center of mass. The paragraph also introduces the concept of moment of inertia and how it is calculated with respect to different axes, emphasizing the symmetry and unified approach in three dimensions.

This section focuses on the calculation of the moment of inertia of a solid using triple integrals. The lecturer provides an example of a solid cone and explains how to calculate its moment of inertia about the z-axis. The paragraph details the process of setting up the integral in cylindrical coordinates, considering the bounds for r, theta, and z, and the simplification that occurs due to the symmetry of the problem. It concludes with an exercise for students to try setting up the integral in a different order and to practice the calculation.

The final paragraph presents an example of setting up a triple integral for a region where z is greater than one minus y inside the unit ball centered at the origin. The lecturer describes the geometric shape formed by the intersection of the plane and the sphere, which is a slanted circle, and its projection in the xy-plane, which is an ellipse. The paragraph discusses the process of determining the bounds for x and y given y and the complexity involved in setting up the triple integral in rectangular coordinates. It concludes with a hint at the bounds for y and x and encourages students to practice the calculation and understand the derivation of these bounds.