STEM-Master
Calculus IIIChapter 4: Multiple IntegralsSection 4-5

Triple Integrals — Question 3

PDF ↗

Question 3

Find the volume of the first-octant tetrahedron x/2+y/3+z/6≤1x/2+y/3+z/6\le 1.

Tasks

  1. Write bounds in the order dzdydxdz\,dy\,dx.

  2. Evaluate.

  3. Verify geometrically.

Original worksheet page 1: question and worked solution for 4-5-003
Show solutionHide solution

Question 3 – Solution

Strategy. Solve the plane successively for zz and for the projected yy-boundary.

Step 1: GeometryThe coordinate-plane faces meet the sloping face at the three axis intercepts shown.

See the diagram in the original worksheet below.

Step 2: Bounds and integral0≤x≤20\le x\le 2, 0≤y≤3−3x/20\le y\le 3-3x/2, and 0≤z≤6−3x−2y0\le z\le 6-3x-2y. Hence V=∫02∫03−3x/2(6−3x−2y)dydx=∫0294(2−x)2dx=6V=\int_0^2\int_0^{3-3x/2}(6-3x-2y)dy\,dx=\int_0^2\frac 94(2-x)^2dx=\boxed{6}.

VerificationThe intercept formula gives V=(1/6)(2)(3)(6)=6V=(1/6)(2)(3)(6)=6.

Original worksheet page 2: question and worked solution for 4-5-003

Original worksheet layout. Use Enlarge or open the PDF for a closer view.

← Section questionsChapter contents