To change a triple integral into cylindrical coordinates, we’ll need to convert the limits of integration, the function itself, and dV from rectangular coordinates into cylindrical coordinates. The variable z remains, but x will change to rcos(theta), and y will change to rsin(theta). dV will convert to r dz dr d(theta).
Read MoreIterated integrals are double or triple integrals whose limits of integration are already specified. In this lesson, we’ll look at how to evaluate triple iterated integrals by working from the inside toward the outside.
Read MoreWe can integrate in any order, so we’ll try to integrate in whichever order is easiest, depending on the limits of integration, which we’ll find by analyzing the object E. Keep in mind that, when it comes to limits of integration, you want limits that are in terms of the variables left to be integrated.
Read MoreTo find the volume from a triple integral using cylindrical coordinates, we’ll first convert the triple integral from rectangular coordinates into cylindrical coordinates. We’ll need to convert the function, the differentials, and the bounds on each of the three integrals. Once the triple integral is expressed in cylindrical coordinates, then we can integrate to find volume.
Read MoreIn order to use the triple integral average value formula, we’ll have find the volume of the object, plus the domain of x, y, and z so that we can set limits of integration, turn the triple integral into an iterated integral, and replace dV with dzdydx.
Read MoreGiven a region defined in uvw-space, we can use a Jacobian transformation to redefine it in xyz-space, or vice versa. We’ll use a 3x3 determinant formula to calculate the Jacobian.
Read MoreWe can use triple integrals and spherical coordinates to solve for the volume of a solid sphere. To convert from rectangular coordinates to spherical coordinates, we use a set of spherical conversion formulas.
Read MoreSimilarly to the way we used midpoints to approximate single integrals by taking the midpoint at the top of each approximating rectangle, and to the way we used midpoints to approximate double integrals by taking the midpoint at the top of each approximating prism, we can use midpoints to approximate a triple integral by taking the midpoint of each sub-cube.
Read MoreLike cartesian (or rectangular) coordinates and polar coordinates, cylindrical coordinates are just another way to describe points in three-dimensional space. Cylindrical coordinates are exactly the same as polar coordinates, just in three-dimensional space instead of two-dimensional space.
Read MoreThere are six ways to express an iterated triple integral. While the function inside the integral always stays the same, the order of integration will change, and the limits of integration will change to match the order.
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