High School
SAT
ACT
University
MCAT
LSAT
DAT
Wize University Calculus 2 Textbook > Bonus: Integral calculus in several variables (Videos Coming Soon)

Triple Integration (rectangular)

Double Integration (polar)Previous Section
Triple Integration (cylindrical)Next Section
Popular Courses
Find My Course

Triple Integration (rectangular)

Wize Concept
Iterated Triple Integral. Let D⊂R3D \subset \mathbb{R}^3 be defined by
g1(x,y)≤z≤g2(x,y),(x,y)∈Dxy,g_1(x, y) \le z \le g_2(x, y),\quad (x, y) \in D_{xy},
where g1(x,y)g_1(x, y) and g2(x,y)g_2(x, y) are continuous on DxyD_{xy}, and DxyD_{xy} is a closed and bounded set in R2\mathbb{R}^2 whose
boundary is a piecewise smooth curve. If f(x,y,z)f(x, y, z) is continuous on DD, then
∭Df(x,y,z)dV=∬D(∫g1(x,y)g2(x,y)f(x,y,z)dz)dA\iiint\limits_{D}f(x,y,z)dV=\iint\limits_{D}\Bigg(\int_{g_1(x,y)}^{g_2(x,y)}f(x,y,z)dz\Bigg)dA
The 2-fold iterated integral for double integration can then be used to further simplify this integral.
→\to Notes:
  1. The iterated integral can be rearranged in any order (if the bounds and integrand are continuous); take care to change the bounds of integration as well.
  2. If the integrand is can be written as a product of functions, one of which is independent of the other variables, then that integral can be done independently.
  3. The positive octant is the piece of R3\mathbb{R}^3 such that x,y,z≥0x, y, z \ge 0.

Practice: Triple integration (rectangular)

Evaluate the iterated integral
∫−12∫01∫13(y+4xz3)dxdydz\displaystyle \int_{-1}^2\int_0^1\int_1^3(y+4xz^3)dxdydz

Practice: Triple integration (rectangular)

Compute ∭DxdV,\displaystyle\iiint\limits_Dx dV, where D is the region under the plane x + 2y + 3z = 6 that lies in the octant x,y,z≥0.x, y, z\ge 0.