Problem 23 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use triple iterated integrals to find the indicated quantities. Volume of the solid bounded by the cylinders \(x^{2}=y\) and \(z^{2}=y\) and the plane \(y=1\)

Step-by-Step Solution

Answer

The volume of the solid is 2 cubic units.

Step 1: Identify the Region

The solid is bounded by the cylinders given by the equations \(x^2 = y\) and \(z^2 = y\), and the plane \(y = 1\). To describe this solid, notice that both cylinders intersect the plane \(y = 0\), and they extend upwards until the plane \(y = 1\). This defines a volume in the region where \(0 \le y \le 1\), \(-\sqrt{y} \le x \le \sqrt{y}\), and \(-\sqrt{y} \le z \le \sqrt{y}\).

Step 2: Set Up the Integral

To calculate the volume, we need to set up an iterated integral in the order \(dxdzdy\). The limits for \(x\) and \(z\) will depend on \(y\):- For \(x\), the limits are from \(-\sqrt{y}\) to \(\sqrt{y}\).- For \(z\), the limits are from \(-\sqrt{y}\) to \(\sqrt{y}\).- For \(y\), the limits are from 0 to 1.Thus, the integral is:\[\int_{0}^{1} \int_{-\sqrt{y}}^{\sqrt{y}} \int_{-\sqrt{y}}^{\sqrt{y}} 1\, dx \, dz \, dy\]

Step 3: Evaluate the Inner Integral

Start by evaluating the inner integral with respect to \(x\):\[\int_{-\sqrt{y}}^{\sqrt{y}} 1 \, dx = x \Big|_{-\sqrt{y}}^{\sqrt{y}} = (\sqrt{y} - (-\sqrt{y})) = 2\sqrt{y}\]This expression represents the area for the section of the solid at a fixed \(y\) and \(z\).

Step 4: Evaluate the Middle Integral

Now evaluate the resulting integral with respect to \(z\):\[\int_{-\sqrt{y}}^{\sqrt{y}} 2\sqrt{y} \, dz = 2\sqrt{y} \cdot z \Big|_{-\sqrt{y}}^{\sqrt{y}} = 2\sqrt{y} \cdot (\sqrt{y} - (-\sqrt{y})) = 4y\]

Step 5: Evaluate the Outer Integral

Finally, evaluate the last integral with respect to \(y\):\[\int_{0}^{1} 4y \, dy = 4 \cdot \frac{y^2}{2} \Big|_{0}^{1} = 4 \cdot \frac{1}{2} = 2\]This calculation gives the total volume of the solid.

Key Concepts

Iterated Integrals

Iterated integrals are a method of calculating multiple integrals in sequence. In simple terms, these are integral evaluations performed one after the other, each having its own limits.

This technique is particularly useful when calculating volumes or other quantities in multivariable calculus where boundaries vary between each integral.

• The order of integration is pivotal because it dictates which variable is integrated first.
• Each integral's limits depend on the variables integrated before it, capturing the geometry of the region in question.

In the given exercise, the order of integration is \(dxdzdy\). This sequence suggests that we first integrate with respect to \(x\), then \(z\), and finally \(y\). The boundaries for each integral can change based on the dependency on previously integrated variables.

Cylinder Surfaces

Cylinder surfaces in multivariable calculus define regions and surfaces based on given equations. They are not always circular; in our case, they are parabolic in nature.

• The equation \(x^{2}=y\) represents a parabolic cylinder, opening in two dimensions (the \(x\) and \(y\) plane).
• Similarly, \(z^{2}=y\) describes another cylinder, opening in the \(z-y\) plane.

These equations define the edges of the solid volume and are crucial in setting the limits of integration. By intersecting with a plane like \(y=1\), they encapsulate a bounded geometry from which volume calculations can be made using iterated integrals.

Solid Geometry

Understanding solid geometry involves recognizing how three-dimensional shapes are described mathematically. This includes defining solids bounded by surfaces and planes within a given space.

In our task, the shape is determined by two cylinders and a plane.

• These elements together form a unique shape, a shared region where all surfaces intersect.
• It is essential to visualize this in 3D space to set up the integration correctly.

The key is recognizing where these surfaces intersect and using these interactions to define a volume. Here, the intersection of cylinders and the plane results in specific limits for each variable, notably \(x, z\) both bounded by \(\pm\sqrt{y}\) whenever \(0 \leq y \leq 1\). These intersections inform how the iterated integrals should be constructed to capture the bounded volume.

Volume Calculation

Volume calculation using iterated integrals includes integrating over the specific region defined by the solid's boundaries. The process requires setting multiple integral limits appropriately to account for changes in each variable as they relate to each other.

• Begin by evaluating the inner integrals to first capture slices of the volume, like integrating over \(x\) and observing boundary influence from \(y\).
• Continue with the next layer, here with \(z\), before considering the outermost integral over \(y\).

Each step condenses the shape's geometric nature into values reflecting partial volumes, leading toward the full volume when entirely accounted for. In this task, through this layered approach, the final result shows that the volume is \2\ cubic units. This calculated result is the outcome of methodically processing the contributions of the defined geometric boundaries.