Set up the triple integral that gives the volume of the solid tetrahedron formed by the vertices (0, 0, 0), (1, 1, 0), (0, 1, 0), and (0, 1, 1).
Understand the Problem
The question is asking us to set up a triple integral to calculate the volume of a tetrahedron defined by specific vertices. The key concepts involve understanding the geometric representation of the tetrahedron and applying the correct limits for a triple integral.
Answer
The triple integral is \( V = \int_{0}^{1} \int_{x}^{1} \int_{0}^{y} dz \, dy \, dx \).
Answer for screen readers
The triple integral set up to calculate the volume of the tetrahedron is: $$ V = \int_{0}^{1} \int_{x}^{1} \int_{0}^{y} dz , dy , dx $$
Steps to Solve
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Identify the vertices of the tetrahedron
The given vertices are ( (0, 0, 0) ), ( (1, 1, 0) ), ( (0, 1, 0) ), and ( (0, 1, 1) ). Understanding the positions of these points will help in establishing the limits of integration. -
Determine the planes that define the tetrahedron
The tetrahedron lies above the triangle formed by points ( (0, 0, 0) ), ( (1, 1, 0) ), and ( (0, 1, 0) ), and is limited at the top by the plane connecting the fourth vertex ( (0, 1, 1) ). -
Establish the limits for integration
We will integrate in the order ( dz , dy , dx ).
- For a fixed ( x ) in the range from 0 to 1, ( y ) ranges from ( x ) to ( 1 ).
- For a fixed ( y ), ( z ) ranges from ( 0 ) to ( y ).
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Write the triple integral
The triple integral for the volume ( V ) can be set up as follows: $$ V = \int_{0}^{1} \int_{x}^{1} \int_{0}^{y} dz , dy , dx $$ -
Simplifying the integral
Now, we can express this integral in a more standard format (though not always necessary for setting it up): $$ V = \int_{0}^{1} \int_{x}^{1} (y - 0) , dy , dx $$
The triple integral set up to calculate the volume of the tetrahedron is: $$ V = \int_{0}^{1} \int_{x}^{1} \int_{0}^{y} dz , dy , dx $$
More Information
The volume of the tetrahedron can also be calculated using the formula for the volume of a tetrahedron, which is ( \frac{1}{6} ) times the absolute value of the determinant of a matrix formed by its vertices. This integral method provides a way to visualize and compute the volume using calculus.
Tips
- Forgetting to properly establish the limits based on the triangular base and the height of the tetrahedron.
- Mixing the order of integration without adjusting the limits accordingly.
- Overlooking the need for calculating ( z ) limits based on ( y ).
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