Get Triple Integrals

1 Answer. A double integral is used for integrating over a two-dimensional region, while a triple integral is used for integrating over a three-dimensional region. ... Hence, we would integrate over sphere with a double integral, but we would use a triple integral to integrate over the volume that the sphere bounds.

To compute a triple integral we use Fubini's theorem, which states that if f(x,y,z) is continuous on a rectangular box B=[a,b]×[c,d]×[e,f], then Bf(x,y,z)dV= fe dc baf(x,y,z)dxdydz. ... To compute the volume of a general solid bounded region E we use the triple integral V(E)= E1dV.

Just as definite integrals can be used to find the area under a curve, they can also be used to find the area between two curves. To find the area between two curves defined by functions, integrate the difference of the functions. ... The arc length of a curve can be calculated using a definite integral.

The definite integral of 1 is the area of a rectangle between x_lo and x_hi where x_hi > x_lo. In general, the indefinite integral of 1 is not defined, except to an uncertainty of an additive real constant, C. However, in the special case when x_lo = 0, the indefinite integral of 1 is equal to x_hi.

So the integral of 2 can be 2x + 3, 2x + 5, 2x, etc. For this reason, when we integrate, we have to add a constant. So the integral of 2 is 2x + c, where c is a constant. A "S" shaped symbol is used to mean the integral of, and dx is written at the end of the terms to be integrated, meaning "with respect to x".

You can use both double and triple integrals when calculating a volume. Let me explain you using an example for calculating an area, same applies to volume. What you are doing is basically summing infinitely many stripes of length f(x) and base length dx.

f(x)dx=F(x)+C,ifF (x)=f(x). In this definition, the is called the integral symbol, f(x) is called the integrand, x is called the variable of integration, dx is called the differential of the variable x, and C is called the constant of integration.

triple integrals can be used to 1) find volume, just like the double integral, and to 2) find mass, when the volume of the region we're interested in has variable density.

triple integrals can be used to 1) find volume, just like the double integral, and to 2) find mass, when the volume of the region we're interested in has variable density.