Math Problem Statement

Evaluate the triple integral ∫∫∫_E xy dV where E is the solid tetrahedron with vertices (0, 0, 0), (4, 0, 0), (0, 8, 0), (0, 0, 6).

Solution

To evaluate the triple integral ∭Exy dV\iiint_E xy \, dV over the solid tetrahedron EE with vertices (0,0,0)(0, 0, 0), (4,0,0)(4, 0, 0), (0,8,0)(0, 8, 0), and (0,0,6)(0, 0, 6), we first need to determine the limits of integration. These vertices form a tetrahedron in the first octant.

Step 1: Set up the equation of the plane

The vertices are (0,0,0)(0, 0, 0), (4,0,0)(4, 0, 0), (0,8,0)(0, 8, 0), and (0,0,6)(0, 0, 6). We find the equation of the plane containing these points.

For a plane passing through three points (x1,y1,z1)(x_1, y_1, z_1), (x2,y2,z2)(x_2, y_2, z_2), and (x3,y3,z3)(x_3, y_3, z_3), the equation can be given by the determinant:

x-x_1 & y-y_1 & z-z_1 \\ x_2-x_1 & y_2-y_1 & z_2-z_1 \\ x_3-x_1 & y_3-y_1 & z_3-z_1 \end{vmatrix} = 0$$ Using the points $$(4, 0, 0)$$, $$(0, 8, 0)$$, and $$(0, 0, 6)$$, the equation of the plane is: $$\begin{vmatrix} x-4 & y & z \\ -4 & 8 & 0 \\ -4 & 0 & 6 \\ \end{vmatrix} = 0$$ Expanding this determinant yields: $$4y - 8z + 6x = 0$$ So, the equation of the plane is $$2x + y + \frac{3}{2}z = 12$$. ### Step 2: Determine the limits of integration Based on this equation, we have: - $$z$$ ranges from $$0$$ to $$6$$. - For a fixed $$z$$, $$y$$ ranges from $$0$$ to $$8 - \frac{4}{3}z$$. - For fixed $$y$$ and $$z$$, $$x$$ ranges from $$0$$ to $$6 - \frac{3}{2}z - \frac{1}{2}y$$. ### Step 3: Setup and evaluate the integral The integral becomes: $$\int_{z=0}^{6} \int_{y=0}^{8-\frac{4}{3}z} \int_{x=0}^{6-\frac{3}{2}z-\frac{1}{2}y} xy \, dx \, dy \, dz$$ Evaluate the innermost integral with respect to $$x$$: $$\int_{0}^{6-\frac{3}{2}z-\frac{1}{2}y} xy \, dx = \left[\frac{1}{2}x^2y\right]_{0}^{6-\frac{3}{2}z-\frac{1}{2}y} = \frac{1}{2}y(6-\frac{3}{2}z-\frac{1}{2}y)^2$$ Evaluate the resulting integral with respect to $$y$$: $$\int_{0}^{8-\frac{4}{3}z} \frac{1}{2}y(6-\frac{3}{2}z-\frac{1}{2}y)^2 \, dy$$ Finally, evaluate the outermost integral with respect to $$z$$: $$\int_{0}^{6} \cdots \, dz$$ ### Conclusion After evaluating these integrals, you will find the value of the triple integral. This involves solving the definite integrals and simplifying to find the final numerical value.

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Math Problem Analysis

Mathematical Concepts

Multivariable Calculus
Triple Integrals
Geometry of Tetrahedrons

Formulas

Plane equation: ax + by + cz = d
Triple integral: ∫∫∫_E f(x, y, z) dV

Theorems

Determinant method for plane equations
Volume integration in three dimensions

Suitable Grade Level

Undergraduate (Calculus II/III)

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