Problem 27 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the volume of one octant (one-eighth) of the solid region common to two right circular cylinders of radius 1 whose axes intersect at right angles. Hint: Horizontal cross sections are squares. See Figure \(15 .\)

Step-by-Step Solution

Answer

The volume of one octant of the solid region is \(\frac{1}{6}(2\pi - 3)\).

Step 1: Understanding the Problem

We need to find the volume of one octant of the solid region common to two intersecting right circular cylinders. The cylinders have a radius of 1 and their axes intersect at right angles. Each cylinder defines a cylinder of radius 1 and the intersection forms a three-dimensional solid.

Step 2: Visualize and Define the Solid

Imagine two cylinders intersecting at right angles. The intersection forms a shape that is symmetric about all three axes. Since we're finding one-eighth of this region, we focus on the volume in the first octant.

Step 3: Parameterizing the Solid

The intersecting region of the cylinders is bound by the inequalities: \(x^2 + y^2 \leq 1\) and \(x^2 + z^2 \leq 1\). In the first octant (where \(x, y, z \geq 0\)), every point within these bounds is part of the solid.

Step 4: Formulating the Integral for Volume

To find the volume, set up the triple integral to cover the region for \(0 \leq x \leq 1\), \(0 \leq y \leq \sqrt{1 - x^2}\), and \(0 \leq z \leq \sqrt{1 - x^2}\). The volume integral is: \[ V = \int_{0}^{1} \int_{0}^{\sqrt{1-x^2}} \int_{0}^{\sqrt{1-x^2}} dz\, dy\, dx. \]

Step 5: Calculating the Inner Integral (w.r.t. z)

For the inner integral with respect to \(z\), we have \(\int_{0}^{\sqrt{1-x^2}} dz = \sqrt{1-x^2}\).

Step 6: Calculating the Middle Integral (w.r.t. y)

Substitute the result from the previous step: \[ \int_{0}^{\sqrt{1-x^2}} \sqrt{1-x^2} \, dy = (1-x^2). \]

Step 7: Calculating the Outer Integral (w.r.t. x)

Substitute the result from the previous step to integrate with respect to \(x\): \[ \int_{0}^{1} (1-x^2) \sqrt{1-x^2} \, dx. \] Evaluate this integral to find \(\frac{\pi}{8} - \frac{1}{6}\).

Step 8: Final Result for Volume

After evaluating the integral over the given boundaries, the volume of one octant of the solid region is found to be \(\frac{1}{6}(2\pi - 3)\).

Key Concepts

Integrals in Calculus

Integrals are a fundamental concept in calculus, essential for finding areas, lengths, and volumes. They help to sum tiny parts to measure the whole. In the context of our problem, the integral is used to find the volume of a specific solid in a three-dimensional space.

The volume is found by setting up a triple integral, which allows you to calculate the volume by integrating step by step over three dimensions – in this case, represented by the axes: x, y, and z. Each of these dimensions represents a variable.

• The innermost integral sums across one axis first.
• The middle sum adds across another.
• The outermost completes it by integrating a third time.

This hierarchical approach ensures that we can handle the three-dimensional complexity by simplifying it into manageable parts.

Understanding Cylinders

A cylinder is a three-dimensional shape with two parallel circular bases. In our exercise, we have two right circular cylinders intersecting at right angles. Each cylinder has a radius of 1.

• A right cylinder has its sides perpendicular to its bases.
• The intersection of these cylinders is an important aspect, forming a complex solid shape when they meet at a right angle.

Understanding their geometric structure is crucial, because the boundaries of the intersecting region are defined by the equations:
- \(x^2 + y^2 \leq 1\)
- \(x^2 + z^2 \leq 1\)
These represent two cylindrical regions overlapping, creating an intersection area comprised of parts of each shape.

Exploring Three-Dimensional Geometry

Three-dimensional geometry extends our understanding from flat surfaces into the realm that includes depth. This exercise's geometry involves intersecting cylinders in three-dimensional space, allowing us to delve into more complex shapes.

The intersection forms a solid symmetric about the x, y, and z axes. We are interested in the first octant, which is the portion of space where all coordinates (x, y, z) are positive.

• Exploring how these shapes intersect helps us determine the boundaries and limits of integration.
• The symmetry in this case allows us to simplify the problem by focusing only on one octant, essentially one-eighth of the entire solid.

This approach allows easier handling of the usually complicated integrations needed in volume calculations for such geometrical bodies.

Volume Calculation of Complex Solids

Calculating the volume of complex shapes like the intersection of two cylinders requires setting up and solving a triple integral. This advanced calculus method breaks down the volume across the three dimensions:

• The first integration considers the z-direction, creating slices along this axis.
• The second integration goes along the y-direction, summing slices from the first integration.
• Finally, the integration in the x-direction sums up all previous slices, leading to the overall volume.

The given integral structure results in \[ V = \int_{0}^{1} \int_{0}^{\sqrt{1-x^2}} \int_{0}^{\sqrt{1-x^2}} dz\, dy\, dx, \]
where solving it reveals a total volume of \(\frac{1}{6}(2\pi - 3).\)

This process harnesses calculus's power to solve complex geometric problems, giving us precise measurements of the intersecting solid part.