Problem 53 - Calculus Early Transcendentals

Question

Use a triple integral to compute the volume of the following regions. The pyramid with vertices (0,0,0),(2,0,0),(2,2,0),(0,2,0) and (0,0,4)

Step-by-Step Solution

Answer

Answer: The volume of the pyramid is 8 cubic units.

Step 1: Find the equation of the plane containing the vertices

To find the equation of the plane, we can use any three points on the plane that are not collinear. Let's use (0,0,0), (2,0,0), and (0,0,4). The vectors formed by these points are: Vector A = (2,0,0) - (0,0,0) = (2,0,0) Vector B = (0,0,4) - (0,0,0) = (0,0,4) Now we can find the normal vector by taking the cross product of A and B: Normal vector N = A x B = (0,-8,0) The equation for the plane can be written as Ax + By + Cz = D, where (A,B,C) are the components of the normal vector and D can be determined by substituting coordinates from any point in the plane. Let's use (0,0,0): 0x - 8y + 0z = D Substituting the point (0,0,0), we find D = 0. Thus, the equation of the plane is: 0x - 8y + 0z = 0, or y = 0.

Step 2: Set up the triple integral to compute the volume

The triple integral to compute the volume of the pyramid is given by the following expression: $$\int_{x=0}^{2} \int_{y=0}^{2} \int_{z=0}^{4-\frac{4x}{2}} dz\, dy\, dx$$ The limits of integration for x and y are quite straightforward, as they correspond to the coordinates of the vertices of the base of the pyramid. The limits of integration for z are determined by the equation of the plane, which we found in step 1.

Step 3: Evaluate the triple integral to compute the volume

Now, we will evaluate the triple integral: $$\int_{x=0}^{2} \int_{y=0}^{2} \int_{z=0}^{4-\frac{4x}{2}} dz\, dy\, dx = \int_{x=0}^{2} \int_{y=0}^{2} (4-2x) dy\, dx$$ First, we will integrate with respect to z: $$\int_{x=0}^{2} \int_{y=0}^{2} (4-2x) dy\, dx = \int_{x=0}^{2} [(4-2x)y]_{y=0}^{2} dx$$ Next, we will integrate with respect to y: $$\int_{x=0}^{2} [(4-2x)(2)] dx = \int_{x=0}^{2} (8-4x) dx$$ Finally, we will integrate with respect to x: $$[(8-4x)x]_{x=0}^{2} = (8-4(2))(2) = 8$$ The volume of the pyramid is 8 cubic units.

Key Concepts

Volume Calculation

Calculating the volume of a geometric shape using multiple integrals is a fundamental concept in calculus. In the case of a pyramid, a **triple integral** is employed to determine the volume. This method involves integrating a function over a three-dimensional region.

Here’s how we approach it:

• First, identify the region of integration, which represents the space occupied by the pyramid.
• Set up the triple integral with limits based on the geometrical description of the region.
• Integrate systematically, usually starting from the innermost integral to the outermost, representing depth, width, and height.

In this pyramid example, the volume is calculated by evaluating the definite integral through the specified limits of each coordinate. The process provides an exact measure of the space within the pyramid.

Pyramid Geometry

When assessing the geometry of a pyramid, it’s crucial to comprehend the structural arrangement of its vertices. In our example:

• The pyramid base resides in the xy-plane with a rectangular configuration defined by the vertices (0,0,0), (2,0,0), (2,2,0), and (0,2,0).
• The apex of the pyramid is at the point (0,0,4), situated along the z-axis.

These vertices establish a pyramid that rises perpendicularly from its rectangular base. To define the characteristics of the pyramid fully, determining the equation of the plane that includes the apex and one side of the base is necessary. This plane helps define the limits for the vertical integration in the triple integral. Understanding the spatial orientation and boundaries of these geometric elements is key to accurately setting up the integral needed for volume calculation.

Integration Limits

Setting the correct integration limits in a triple integral is essential to solving the volume of a region properly. Each variable in the integral needs its boundary defined.

For the pyramid:

• The limits for **x** range from 0 to 2, corresponding to the length of the base along the x-axis.
• The limits for **y** are similar to those for x, ranging from 0 to 2, depicting the width along the y-axis.
• The variable **z** represents the height and has more complex limits influenced by the shape. Here, it depends on the equation of the plane z = 4 - 2x, establishing limited space that decreases with increasing x.

These limits ensure that the count is restricted to the desired geometry of the pyramid. This configuration captures the sloped boundary set by the triangular faces, delivering the finite integration domain necessary for accurate volume computation.