Question

Sketch the curved wedge bounded by the surfaces \(y^{2}=4 a x, x+z=a\) and \(z=0\), and hence calculate its volume \(V\).

Short answer

Use triple integration in region \(0 \leq x\leq a,z\ and 0 \leq z+x= a\). Evaluate with bounds.

Step-by-step solution

Identify the surfaces and region

The given surfaces are described by the equations \(y^{2}=4ax\), \(x+z=a\), and \(z=0\). The surface \(y^{2}=4ax\) is a parabolic cylinder, \(x+z=a\) is a plane, and \(z=0\) is the \(xy\)-plane.

Find intersection curves

To understand the boundaries, first find where the plane \(x+z=a\) intersects the parabolic cylinder \(y^{2}=4ax\). Substituting \(z=0\) into the plane's equation gives \(x = a\), and substituting \(x+z=a\) into the parabolic cylinder gives the curve in the \(xz\)-plane, with \(z=a-x\).

Set up the volume integral

To compute the volume bounded by these surfaces, integrate over the region in the \(xz\)-plane, from \(x=0\) to \(x=a\) and \(z=0\) to \(z=a-x\). Integrate the height of the region, given by \(y\)'s bounds from the parabolic cylinder.

Evaluate inner integral

Since the parabolic cylinder is given by \(y^{2}=4ax\), we solve for \(y\) to get the limits: \(-2\root ax \root to 2\root ax\). Integrate with respect to \(y\), giving \(2 \root y^{2}=4ax\).

Evaluate outer integrals

First integrate the expression \(2\root y^{2}=4ax\)\ with respect to y, then integrate with respect to \(x\) from \(0\) to \(a\) and with respect to \(z\) from \(0\) to \(a-x\). Combine results to get final volume.

Key concepts

Volume Integrals

Volume integrals are used to find the volume of three-dimensional regions. In problems involving multiple variables, we often integrate by layers, summing up small volumes. We start by setting up an integral over the region of interest. It's crucial to understand the shape and boundaries of this region.
To calculate these integrals, we usually proceed in steps:

• Identify the region and surfaces involved
• Determine the limits of integration
• Set up and evaluate the integral, often broken into an inner and outer part

Understanding volume integrals is key in multivariable calculus.

Parabolic Cylinder

A parabolic cylinder is a surface defined by a parabola that extends infinitely along one axis. In this problem, it is given by the equation: \( y^2 = 4ax \). This shows that for any slice perpendicular to the x-axis, the curve follows a parabola.
Key features to note:

• It extends infinitely along the z-axis
• Any cross-section parallel to the yz-plane is a parabola
• Vertices of these parabolas lie on the x-axis

Recognizing the shape and orientation of a parabolic cylinder helps in visualizing and setting up the bounds for integration.

Plane Intersections

Plane intersections occur where two or more surfaces meet. For this exercise, the plane \( x + z = a \) intersects the parabolic cylinder \( y^2 = 4ax \).
By substituting \( z = 0 \) in the plane equation, we first find:

• The intersection line on the x-axis: \( x = a \)

Next, substituting \( x + z = a \) into \( y^2 = 4ax \) helps identify the curve of intersection. These intersections are crucial for understanding the boundary of the region over which to integrate, informing our limits.

Limits of Integration

Determining limits of integration is fundamental to solving volume integrals. We need to find these limits based on the intersections and geometry of the region. For this problem:

• The x-axis limits are from \( 0 \) to \( a \)
• The z-axis limits are from \( 0 \) to \( a - x \)

For the y-axis, based on the parabolic cylinder, the bounds are: \( -\text{2}\root {4ax} \) to \( \text{2}\root {4ax} \).
Setting these limits accurately is necessary for correctly evaluating the volume integral in multivariable calculus. Begin by integrating over the innermost variable first, then move outward.