Evaluate The Triple Integral Exdv Where E Is The Solid Bounded By The Paraboloid X=10y² + 10z² And X=10.
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Introduction
In multivariable calculus, the evaluation of triple integrals plays a crucial role in understanding the volume and other physical properties of three-dimensional regions. The problem at hand involves computing the triple integral \(\iiint_E x \, dv\), where \(E\) is a solid bounded by the paraboloid \(x=10y^2 + 10z^2\) and the plane \(x=10\). This is a classic example of integrating over a solid region bounded by quadratic surfaces, which often requires converting the integral into a more convenient coordinate system, such as cylindrical or spherical coordinates, to simplify the process.
Understanding how to evaluate such integrals is fundamental in applications across physics, engineering, and mathematics. These include calculating mass, center of mass, moments of inertia, and other physical quantities where the region of integration is defined by quadratic boundaries.
In this article, we will explore the steps involved in setting up and evaluating the triple integral over the specified region. We will analyze the geometric shape of the solid, choose an appropriate coordinate system, and perform the integration systematically to arrive at the final result. This example not only demonstrates the calculation process but also highlights important techniques in multivariable calculus.
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Geometric Description of the Region \(E\)
The Boundaries
The region \(E\) is bounded by:
- The paraboloid \(x=10y^2 + 10z^2\)
- The plane \(x=10\)
These surfaces intersect to form the solid region of integration.
Visualizing the Region
To better understand the shape, consider the following:
- The paraboloid opens along the positive \(x\)-axis since \(x=10(y^2 + z^2)\).
- For \(x=0\), the paraboloid passes through the origin.
- As \(x\) increases, the cross-section of the paraboloid in the \(yz\)-plane is a circle of radius \(\sqrt{x/10}\).
The plane \(x=10\) intersects the paraboloid where:
\[
10 = 10(y^2 + z^2) \Rightarrow y^2 + z^2 = 1
\]
This indicates that the intersection is a circle of radius 1 in the \(yz\)-plane at \(x=10\).
Summary of the Solid \(E\)
- The region starts from the vertex at the origin, extending along the \(x\)-axis up to \(x=10\).
- For each fixed \(x\) in \([0, 10]\), the cross-section in the \(yz\)-plane is a circle:
\[
y^2 + z^2 \leq \frac{x}{10}
\]
This description suggests the region is a paraboloid "filling up" from the origin to the plane \(x=10\).
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Choosing the Coordinate System for Integration
Given the symmetry and the nature of the boundaries, cylindrical coordinates are well-suited for integrating over the region \(E\).
Cylindrical Coordinates
Recall the transformations:
\[
\begin{cases}
y = r \cos \theta \\
z = r \sin \theta \\
dv = dx \, dy \, dz = dx \, r \, dr \, d\theta
\end{cases}
\]
The variables range as follows:
- \(r \geq 0\), \(\theta \in [0, 2\pi)\)
- \(x\) varies from 0 to 10
- For each fixed \(x\), the cross-section in the \(yz\)-plane is \(r \leq \sqrt{\frac{x}{10}}\)
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Setting Up the Integral
The integrand is \(x\), and the differential volume element in cylindrical coordinates is \(dv = dx \, r \, dr \, d\theta\).
The limits are:
- \(x: 0 \to 10\)
- \(\theta: 0 \to 2\pi\)
- \(r: 0 \to \sqrt{\frac{x}{10}}\)
Thus, the triple integral becomes:
\[
\iiintE x \, dv = \int{x=0}^{10} \int{\theta=0}^{2\pi} \int{r=0}^{\sqrt{\frac{x}{10}}} x \, r \, dr \, d\theta \, dx
\]
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Evaluating the Integral
Step 1: Integrate with respect to \(r\)
\[
\int0^{\sqrt{\frac{x}{10}}} x \, r \, dr = x \int0^{\sqrt{\frac{x}{10}}} r \, dr = x \left[ \frac{r^2}{2} \right]_0^{\sqrt{\frac{x}{10}}} = x \times \frac{1}{2} \times \left( \frac{x}{10} \right) = \frac{x^2}{20}
\]
Step 2: Integrate with respect to \(\theta\)
\[
\int_0^{2\pi} d\theta = 2\pi
\]
Step 3: Integrate with respect to \(x\)
Now, the integral reduces to:
\[
\iiintE x \, dv = \int0^{10} \left( \frac{x^2}{20} \times 2\pi \right) dx = \frac{2\pi}{20} \int0^{10} x^2 \, dx = \frac{\pi}{10} \left[ \frac{x^3}{3} \right]0^{10} = \frac{\pi}{10} \times \frac{1000}{3} = \frac{\pi}{10} \times \frac{1000}{3}
\]
Simplify:
\[
\frac{\pi}{10} \times \frac{1000}{3} = \frac{1000 \pi}{30} = \frac{100 \pi}{3}
\]
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Final Result
\[
\boxed{
\iiint_E x \, dv = \frac{100 \pi}{3}
}
\]
This is the value of the triple integral of \(x\) over the solid \(E\) bounded by the paraboloid \(x=10y^2 + 10z^2\) and the plane \(x=10\).
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Additional Insights and Applications
Physical Interpretations
Evaluating integrals like \(\iiint_E x \, dv\) can be interpreted as calculating the first moment of the region \(E\) about the \(yz\)-plane, which relates to the center of mass when considering a uniform density distribution.
Variations and Generalizations
- Changing the function \(x\) to other functions allows for computing different physical quantities.
- Modifying the bounds or the surfaces can model different physical or geometric scenarios.
- Extending the method to more complex surfaces involves similar steps, often requiring coordinate transformations to simplify the boundaries.
Importance in Engineering and Physics
Such integrals are fundamental in:
- Calculating mass, centroid, and moments of inertia of 3D objects.
- Analyzing heat distribution within a solid.
- Determining charge or mass distribution in physics.
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Conclusion
The evaluation of the triple integral \(\iiint_E x \, dv\) over the solid bounded by the paraboloid \(x=10y^2 + 10z^2\) and the plane \(x=10\) exemplifies the power of coordinate transformations and methodical setup in multivariable calculus. By recognizing the symmetry and the geometric shape, selecting the cylindrical coordinate system simplifies the integration process significantly. The final result, \(\frac{100 \pi}{3}\), provides a precise measure related to the volume and distribution within this paraboloidal solid.
Understanding and mastering such techniques are essential for tackling complex volume integrals in various scientific and engineering fields, enabling precise analysis and problem-solving in three-dimensional spaces.
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