Problem 24 - Calculus Early Transcendentals

Question

Set up, but do not evaluate, two different iterated integrals equal to the given integral. \(\iint_{\sigma} x^{2} y d S,\) where \(\sigma\) is the portion of the cylinder \(y^{2}+z^{2}=a^{2}\) in the first octant between the planes \(x=0, x=9, z=y,\) and \(z=2 y\)

Step-by-Step Solution

Answer

Two set iterated integrals are \( \int_0^9 \int_{\frac{z}{2}}^{z} \int_{\sqrt{a^2 - z^2}}^{a} x^2 y \; dz \; dy \; dx \) and \( \int_0^9 \int_{z}^{\frac{z}{2}} \int_{0}^{\sqrt{a^2 - y^2}} x^2 y \; dx \; dy \; dz \).

Step 1: Understand the Surface

The surface \( \sigma \) is the portion of the cylinder defined by \( y^2 + z^2 = a^2 \) in the first octant. This means we are considering only the positive values of \( x \), \( y \), and \( z \), within the given bounds. The bounds \( z = y \) and \( z = 2y \) dictate how \( z \) relates to \( y \) when setting up our integrals.

Step 2: Express in Cylindrical Coordinates

Convert the problem into cylindrical coordinates, where \( y = r \sin(\theta) \) and \( z = r \cos(\theta) \). The equation \( y^2 + z^2 = a^2 \) simplifies to \( r = a \). For the first octant: \( 0 \leq \theta \leq \frac{\pi}{2} \). The cylinder's portion is essentially a constant \( z-y \) and \( z-2y \) in terms of angles and \( r \).

Step 3: Set Up the First Iterated Integral

Using order \( dx \; dy \; dz \), we have bounds for \( x \) as \( 0 \leq x \leq 9 \). For each \( x \), \( y \) is bounded by the planes \( \frac{z}{2} \leq y \leq z \). Since \( y^2 + z^2 = a^2 \), the bounds are \( \sqrt{a^2 - z^2} \leq z \leq \min(a, 2y) \). The integral becomes:\[ \int_0^9 \int_{\frac{z}{2}}^{z} \int_{\sqrt{a^2 - z^2}}^{a} x^2 y \; dz \; dy \; dx \]

Step 4: Set Up the Second Iterated Integral

In the second order \( dz \; dy \; dx \), the boundaries switch such that \( 0 \leq y \leq \frac{z}{2} \). \( z \) changes from \( 0 \) to \( y \). Here, the order of integration results in:\[ \int_0^9 \int_{z}^{\min(\frac{z}{2}, a)} \int_{0}^{\sqrt{a^2 - y^2}} x^2 y \; dx \; dy \; dz \]

Step 5: Verify Limits and Function Inside the Integral

Ensure that the function \( x^2 y \) and integration limits match the region described by the intersection of the geometric planes considering that the cylinder is in the first octant.

Key Concepts

Cylindrical Coordinates

Cylindrical coordinates provide a way to represent points in three-dimensional space using a combination of circular and linear references. They can be seen as an extension of polar coordinates.
- **Representation**: A point in cylindrical coordinates is expressed as \((r, \theta, z)\), where: - \(r\) is the radial distance from the origin to the projection of the point in the xy-plane. - \(\theta\) is the angle between the positive x-axis and the line segment from the origin to the point's projection in the xy-plane. - \(z\) is the height of the point above the xy-plane.Using these, the Cartesian coordinates can be transformed as: - \(x = r\cos(\theta)\) - \(y = r\sin(\theta)\) - \(z = z\)In cylindrical coordinates, cylindrical surfaces like the one described by \(y^2 + z^2 = a^2\) become much simpler. This transformation helps reduce the complexity of iterated integrals by aligning more directly with the geometry of cylinders.

Integration Limits

When setting up iterated integrals, correctly understanding the integration limits is crucial. Integration limits define the boundaries of the region over which we're integrating a function. These boundaries are set based on the physical or geometric constraints described in the problem. For our integral \(\begin{bmatrix}0 \leq x \leq 9 \end{bmatrix}\), \(x\) is integrated from 0 to 9, reflecting its constraints between two planes. Therefore, integrating with respect to \(x\) is straightforward.

For \(y\) and \(z\), the conditions given by the surface equation \(y^2 + z^2 = a^2\) add more complexity. For \(y\), the integration is influenced by the boundaries \(z = y\) and \(z = 2y\), dictating that \(y\) can go from \(\frac{z}{2}\) to \(z\). To prevent exceeding the cylinder's limit, \(z\) is constrained between \(\sqrt{a^2 - z^2}\) and \(a\), ensuring the integral remains on the surface of the cylinder.Thus, determining integration limits involves carefully interpreting each constraint and its impact on other variables in the integrals.

Integral Bounds

Integral bounds are a pivotal concept when working with iterated integrals, as they shape the region of integration. They are defined by both the explicit numerical limits provided and the implicit bounds from geometric relationships. In our case, the bounds for the integrals need to encompass both the portion of the cylinder within the given region and the modulation introduced by the plane equations. While the problem's constraints like \(x = 0\) to \(x = 9\) are straightforward, the bounds for \(y\) and \(z\) depend on variable interrelations.

To set integral bounds for iterated integrals:- We start by defining the range for one variable, such as \(x\) being from 0 to 9.- Next, \(y\) and \(z\) are considered within bounding surfaces \(z = y\) and \(z = 2y\) adjusting these bounds based on the octant restrictions, \(z = 0\) to \(\min(a, 2y)\), and respecting the cylinder \(y^2 + z^2 = a^2\).Careful consideration of these bounds ensures that the iterated integral correctly represents the specified geometric region, facilitating an accurate and comprehensible solution.