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A triple integral is used to evaluate a function of three variables over a three-dimensional region, such as finding the volume under a surface.Example Problem
Evaluate the triple integral:\[ \iiint\limits_E 6z\^2 \, dV \]where \(E\) is the region below the plane \(4x + y + 2z = 10\) in the first octant.Steps to Solve
- Identify the Region: The first step is to determine the limits of integration based on the given plane and the first octant constraints (where \(x, y, z \geq 0\)).
- Find the Intersection Points: To find the limits for \(z\), set \(x = 0\) and \(y = 0\) in the plane equation:
- When \(x = 0\): \(y + 2z = 10 \Rightarrow z = 5 - \frac{y}{2}\)
- When \(y = 0\): \(4x + 2z = 10 \Rightarrow z = 5 - 2x\)
- Set Up the Integral: The limits for \(z\) will be from \(0\) to \(\frac{10 - 4x - y}{2}\). The limits for \(y\) will be from \(0\) to \(10 - 4x\), and \(x\) will range from \(0\) to \(2.5\) (where \(4x = 10\)).
- Thus, the integral becomes:
\[ \int_0\^{2.5} \int_0\^{10 - 4x} \int_0\^{\frac{10 - 4x - y}{2}} 6z\^2 \, dz \, dy \, dx \]- Evaluate the Integral:
- First, integrate with respect to \(z\):
\[ \int 6z\^2 \, dz = 2z\^3 \Big|_0\^{\frac{10 - 4x - y}{2}} = 2\left(\frac{10 - 4x - y}{2}\right)\^3 \]- Substitute this back into the integral and continue integrating with respect to \(y\) and then \(x\).
- Final Calculation: After performing the integrations step by step, you will arrive at the final volume or value of the triple integral.
Conclusion
Lamar UniversityCalculus III - Triple Integrals (Practice Problems)Evaluate \ ( \displaystyle \iiint\limits_ {E} { {6 {z^2}\,dV}}\) where \ (E\) is the region below \ (4x + y + 2z = 10\) in the first octant. Solution Evaluate \ ( \displaystyle \ii…https://tutorial.math.lamar.edu › Problems › CalcIII › TripleIntegrals.aspxMath InsightTriple integral examples - Math InsightSet up the integral of f(x, y, z) f (x, y, z) over W W, the solid “ice cream cone” bounded by the cone z = x2 +y2− −−−−−√ z = x 2 + y 2 and the half-sphere z = 1 −x2 −y2− −−−−−−−−√…https://www.mathinsight.org › triple_integral_examples Calculus III - Triple Integrals - Pauls Online Math Notes
Dec 5, 2025 · In this section we will define the triple integral. We will also illustrate quite a few examples of setting up the limits of integration from the three dimensional region of integration.
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I want to compute one of the most important triple integrals in physics-"the gravitational attraction of a solid sphere." For some reason Isaac Newton had trouble with this integral.
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Triple integral examples - Math Insight
- Set up the integral of f(x,y,z)f(x,y,z) over WW, the solid “ice cream cone”bounded by the cone z=√x2+y2z=x2+y2−−−−−−√ and the half-sphere z=√1−x2−y2, pictured below. Solution: We'll use the shadow method to set up the bounds on the integral. This means we'll write the triple integral as a double integral on the outside and a single integral on the ...
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To get a better understanding of triple integrals let us consider the following example where the triple integral arises in the computation of mass. …
Finding the volume of the solid region bound by the three cylinders x2 + y2 = 1, x2 + z2 = 1 and y2 + z2 = 1 is one of the most famous volume integration problems going back to Archimedes.
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Jan 14, 2025 · Now that we have developed the concept of the triple integral, we need to know how to compute it. Just as in the case of the double …
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Nov 16, 2022 · Here is a set of practice problems to accompany the Triple Integrals section of the Multiple Integrals chapter of the notes for Paul Dawkins Calculus III course at Lamar University.
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