Problem 36 - Thomas Calculus

Question

The region bounded in back by the plane \(x=0,\) on the front and sides by the parabolic cylinder \(x=1-y^{2},\) on the top by the paraboloid \(z=x^{2}+y^{2},\) and on the bottom by the \(x y\) -plane

Step-by-Step Solution

Answer

The volume of the region is \(\frac{16}{15}\).

Step 1: Understand the Boundaries

The region is bounded in the back by the plane \(x=0\) and in the front and on the sides by the parabolic cylinder \(x=1-y^2\). The top of the region is enclosed by the paraboloid \(z=x^2+y^2\), while the bottom is the \(xy\)-plane where \(z=0\). This sets the conditions \(0 \leq x \leq 1-y^2\), and \(0 \leq z \leq x^2+y^2\).

Step 2: Determine the Integration Region for y

Since the parabolic cylinder \(x=1-y^2\) determines the boundary for \(x\), solve for \(y\) when \(x=0\) and \(x=1\). Setting \(x=1-y^2=0\), solve for \(y\), resulting in \(y^2=1\), so \(-1 \leq y \leq 1\).

Step 3: Establish Limits for x

Within the limits of \(y\) found in Step 2, the bounds for \(x\) are determined by the back plane \(x=0\) and the parabolic cylinder \(x=1-y^2\). Thus, given \(y\), \(x\) varies from \(0\) to \(1-y^2\).

Step 4: Set Limits for z

In this case, the variable \(z\) is defined from the \(xy\)-plane at \(z=0\) up to the paraboloid \(z=x^2+y^2\). Therefore, \(z\) varies from \(0\) to \(x^2+y^2\).

Step 5: Integrate to Find the Volume

The volume of the region is given by the triple integral \(\int_{y=-1}^{1}\int_{x=0}^{1-y^2}\int_{z=0}^{x^2+y^2} \, dz \, dx \, dy\). Compute the innermost integral first, which is with respect to \(z\), resulting in \(z\) evaluated from \(0\) to \(x^2+y^2\), yielding \(x^2+y^2\). Then evaluate the integral with respect to \(x\) from \(0\) to \(1-y^2\). Finally, complete the integration by evaluating the integral with respect to \(y\) from \(-1\) to \(1\).

Step 6: Final Integration and Calculation

Evaluate the remaining integrals to compute the volume. Substituting the limits and performing the integration finally gives the volume of the region, which simplifies to \(\frac{16}{15}\).

Key Concepts

Volume Calculation

Understanding how to calculate volume using triple integrals can be a powerful tool in calculus. We use a triple integral because the volume is a three-dimensional measure. This means you'll integrate once for each spatial dimension: length, width, and height. In our problem, the volume of the region is defined using the triple integral \[ \int_{y=-1}^{1}\int_{x=0}^{1-y^2}\int_{z=0}^{x^2+y^2} \, dz \, dx \, dy\].
This expression tells us to first integrate in the direction of the smallest unit of height for each small slice of the region (represented by the variable \(z\)), then horizontally along the \(x\)-axis, and finally along the vertical \(y\)-axis. With each integration, we gather a layer of volume calculations, eventually culminating in the complete volume of the space.
This specific triple integral illustrates how we can calculate volumes of complex regions using calculus, taking into account all spatial boundaries involved.

Paraboloid

A paraboloid is a three-dimensional surface that is shaped like a parabola extended along an axis. Think of it like a stretched or rotated bowl. In this problem, our paraboloid is given by the equation \[ z = x^2 + y^2 \].
This describes a surface that opens upwards, with the vertex or lowest point at the origin \((0, 0, 0)\). Any point on the paraboloid has its \(z\)-coordinate determined by the sum of the squares of its \(x\) and \(y\) coordinates.

• It's the top boundary of the region we are analyzing, limiting how high \(z\) can go.
• This means as \(x\) and \(y\) increase, the height \(z\) increases, giving us a curved upper boundary for the volume we are calculating.

Understanding the shape and equation of the paraboloid helps us recognize how it can serve as a limit to the region's height in this volume problem.

Parabolic Cylinder

A parabolic cylinder is a cylinder whose cross sections are parabolas, rather than circles as in a typical circular cylinder. In our problem, the parabolic cylinder is represented by the equation \[ x = 1 - y^2 \].
This tells us that for each value of \(y\), \(x\) is bound by a parabolic constraint, which restricts the region's width at a given point to a specific horizontal boundary.

• The rule \(x = 1 - y^2\) helps define the side boundary of our volume, showing a restriction parallel to the \(x\)-axis.
• At \(y = 0\), \(x\) reaches its maximum value of 1.
• As \(y\) moves away from 0 to -1 or 1, the width \(x\) narrows down to 0, creating a funnel-like wrap, providing the radius at different points along the \(y\)-axis.

Grasping a parabolic cylinder's role is crucial for setting limits on \(x\), ensuring the volume calculated pertains only to this bounded region.

Integration Limits

Integration limits are crucial in defining the bounds over which we compute integrals. They dictate the scope and range over which we sum up the values in an integral, essentially establishing the region over which we calculate volume.
In our problem, limits are set for each variable:

• For \(x\), limits are from \(0\) to \(1 - y^2\).
• For \(y\), limits are set from \(-1\) to \(1\).
• For \(z\), limits are from \(0\) to \(x^2 + y^2\).

These limits ensure that we only calculate the volume within the specified boundaries:- \(x\) varies according to the parabolic cylinder constraint- \(y\) varies over the full extent where the cylinder and paraboloid define the space- \(z\) is restricted from the base plane up to the curved surface of the paraboloid
Clear understanding of these integration limits allows us to accurately compute the regions of interest when solving volume problems in triple integral calculus.