Find y dV, where E is the solid bounded by the parabolic cylinder z = a and the planes y = 0 and z = 14 – 2y
Arc Length
Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic path. Alternatively, if a curve on a map represents a road, we might want to know how far we need to drive to get to our destination. The distance between two points along a curve is known as arc length.
Line Integral
A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along a curve is very important in mathematics. Usually, by a line integral, we compute the area of the function along the curve. This integral is also known as curvilinear, curve, or path integral in short. If line integrals are to be calculated in the complex plane, then the term contour integral can be used as well.
Triple Integral
Examples:
![### Problem Statement
Find \(\iiint_E y \, dV\), where \(E\) is the solid bounded by the parabolic cylinder \(z = x^2\) and the planes \(y = 0\) and \(z = 14 - 2y\).
### Explanation
This problem involves calculating a triple integral over a specific region \(E\) in three-dimensional space. The region \(E\) is bounded by:
- A parabolic cylinder described by the equation \(z = x^2\)
- The plane \(y = 0\)
- The plane \(z = 14 - 2y\)
To solve this, one needs to set up the limits of integration for the triple integral over \(E\) accordingly.
### Step-by-Step Solution
1. **Identify the bounds for \(y\)**:
Since \(y = 0\) and we need \(z = 14 - 2y\) to encompass the surface, we solve:
\[
z = 14 - 2y \implies 0 \leq y \leq 7
\]
2. **Identify the bounds for \(x\)**:
The cylinder \(z = x^2\) lets us relate \(x\) and \(z\). Over the range of \(z\) from \(x^2\) to \(14 - 2y\), the bounds for \(x\) are identified:
\[
-\sqrt{z} \leq x \leq \sqrt{z}
\]
3. **Identify the bounds for \(z\)**:
For the given \(y\), the \(z\) ranges from the surface of the cylinder upto the plane:
\[
x^2 \leq z \leq 14 - 2y
\]
### Setting Up the Integral
The integral \(\iiint_E y \, dV\) can be set up with the respective bounds:
\[
\int_0^7 \int_{-\sqrt{14 - 2y}}^{\sqrt{14 - 2y}} \int_{x^2}^{14 - 2y} y \, dz \, dx \, dy
\]
The integral needs to be evaluated to get the desired result.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F086767bd-31d2-4529-b304-0e1d54a1ff2d%2Ff26cc973-643c-4eb0-9458-ce9e07dc16a8%2Fn0u6etu_processed.png&w=3840&q=75)
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