Question

Sketch the indicated solid. Then find its volume by an iterated integration. Solid in the first octant bounded by the circular cylinders \(x^{2}+z^{2}=16\) and \(y^{2}+z^{2}=16\) and the coordinate planes

Short answer

The volume is determined by evaluating the iterated integral set for the bounded region in the first octant.

Step-by-step solution

Understand the Boundaries of the Solid

The problem requires analyzing two circular cylinders with radius 4. The first cylinder is defined by the equation \(x^2 + z^2 = 16\), and the second by \(y^2 + z^2 = 16\). Both cylinders intersect the coordinate planes in the first octant where \(x, y,\) and \(z\) are all non-negative.

Identify the Region of Integration

The cylinders restrict their boundaries in the first octant to \(0 \leq x \leq 4\) and \(0 \leq y \leq 4\). The intersection of cylinders and coordinate planes bounds the solid to the region where both cylinder equations are satisfied.

Determine the Integration Limits

Since the integration is over three dimensions, we set up the triple integral for volume. The limits for \(z\) are determined by solving both cylinder equations for \(z\): \(-\sqrt{16 - x^2}\) to \(\sqrt{16 - x^2}\) and \(-\sqrt{16 - y^2}\) to \(\sqrt{16 - y^2}\). However, only non-negative \(z\) values (from \(z=0\) to \(z=\min(\sqrt{16-x^2}, \sqrt{16-y^2})\)) contribute to the volume in the first octant.

Set Up the Triple Integral

Express the volume as a triple integral: \[ V = \int_{x=0}^{4} \int_{y=0}^{4} \int_{z=0}^{\min(\sqrt{16-x^2}, \sqrt{16-y^2})} dz \ dy \ dx \]

Solve the Innermost Integral

The innermost integral with respect to \(z\) is straightforward: \[ \int_{z=0}^{\min(\sqrt{16-x^2}, \sqrt{16-y^2})} dz = \min(\sqrt{16-x^2}, \sqrt{16-y^2}) \]

Simplify and Evaluate the Outer Integrals

Substitute the solution from the innermost integral back into the equation and simplify the double integral. You may notice symmetry that simplifies the calculation. Evaluate: \[ V = \int_{x=0}^{4} \int_{y=0}^{4} \min(\sqrt{16-x^2}, \sqrt{16-y^2}) \, dy \, dx \] and solve the integral by considering sections where \(\sqrt{16-x^2} \leq \sqrt{16-y^2}\) and vice versa.

Compute the Final Volume

After considering all symmetrical sections, compute the integrals to find the total volume of the solid.

Key concepts

Triple Integral

A triple integral is a method used to calculate volumes in three-dimensional space. It takes the integration process a step further than double integrals, which calculate area, by integrating over three variables. In this exercise, the triple integral helps calculate the volume of a solid bounded by specific surfaces or curves in three-dimensional Cartesian space. When dealing with triple integrals, the domain of integration must be carefully defined to reflect the boundaries of the solid. To compute our volume, we need to perform three integrations: one with respect to each of the variables involved - typically denoted as \( x \), \( y \), and \( z \). Performing a triple integral for a volume calculation involves setting up nested integrals, or an 'iterated integral', such as \[ V = \int_{a}^{b} \int_{c}^{d} \int_{e}^{f} dz dy dx \] where the limits \( a, b, c, d, e, \) and \( f \) are defined by the boundaries of the solid.

Cylindrical Coordinates

Cylindrical coordinates are a useful alternative to Cartesian coordinates for solving problems with circular symmetry. This system utilizes the variables \( r \), \( \theta \), and \( z \). The variable \( r \) represents the radius from the origin to the point's projection onto the \( xy \)-plane, \( \theta \) is the angle relative to the \( x \)-axis, and \( z \) remains the same as in Cartesian coordinates. The exercise at hand involves cylindrical structures, making cylindrical coordinates particularly handy. In cylindrical coordinates, the equations \( x^2 + z^2 = 16 \) and \( y^2 + z^2 = 16 \) represent circular cylinders of radius 4, oriented along respective axes. The use of these coordinates can often simplify the integration process by aligning the equations with the symmetry of the shape.

Volume Calculation

Calculating the volume of a solid by integration involves finding the sum of infinitely many infinitesimally small 'slices' of volume. In the case of this exercise, our goal is to find the volume of the solid defined by two intersecting cylinders confined to the first octant.Here's how we set up the integral for this volume:- Identify the boundaries for integration based on geometric constraints.- Solve the equations of the cylinders for \( z \) to establish bounds between \( 0 \) and \( \sqrt{16-x^2} \) or \( \sqrt{16-y^2} \).- Integrate first with respect to \( z \), then \( y \), and finally \( x \).Volume can be expressed in the integral as:\[ V = \int_{0}^{4} \int_{0}^{4} \int_{0}^{\min(\sqrt{16-x^2}, \sqrt{16-y^2})} dz \, dy \, dx \]This expresses the three-dimensional nature of the solid we are considering.

First Octant

The first octant in three-dimensional space refers to the region where all three Cartesian coordinates - \( x \), \( y \), and \( z \) - are non-negative. Imagine the coordinate plane divided into sections like a three-dimensional pie, and the first octant is the first 'slice'.When setting up integrals for volume in this region, it is critical to ensure that conditions \( x \geq 0 \), \( y \geq 0 \), and \( z \geq 0 \) are maintained. In our problem, this means ensuring that the bounds for \( x \) and \( y \) both start at \( 0 \) and end at \( 4 \), reflecting the intersections of the cylinders and coordinate planes only within the first octant.

Coordinate Planes

In three-dimensional geometry, the three "coordinate planes" are the \( xy \)-plane, \( yz \)-plane, and \( xz \)-plane. Each plane is defined by setting one of the spatial dimensions to zero:

• The \( xy \)-plane where \( z \) is zero
• The \( xz \)-plane where \( y \) is zero
• The \( yz \)-plane where \( x \) is zero

These planes form the boundaries of the space in which our solid resides. For our specific problem, the coordinate planes help determine the limits of integration since the solid is confined to the first octant. Understanding these boundaries ensures that the integration considers only the region of interest. By incorporating boundaries given by the equations of the cylinders and the restrictive conditions of the coordinate planes, the integrated volume calculation aligns accurately with the real-world geometrical constraints of the shape.