Preview Activity 12.6.1.
Consider a solid piece of granite in the shape of a box \(B = \{(x,y,z) : 0 \leq x \leq 4, 0 \leq y \leq 6, 0 \leq z \leq 8\}\text{,}\) whose density varies from point to point. Let \(\delta(x, y, z)\) represent the mass density of the piece of granite at point \((x,y,z)\) in kilograms per cubic meter (so we are measuring \(x\text{,}\) \(y\text{,}\) and \(z\) in meters). Our goal is to find the mass of this solid.
Recall that if the density was constant, we could find the mass by multiplying the density and volume; since the density varies from point to point, we will use the approach we did with two-variable lamina problems; We will apply the first step of the classic calculus approach by slicing the solid into smaller pieces on which the density is roughly constant and use these pieces to approximate the total volume.
(a)
For our approximation using smaller pieces, we will partition the interval \([0,4]\) into 2 subintervals of equal length, the interval \([0,6]\) into 3 subintervals of equal length, and the interval \([0,8]\) into 2 subintervals of equal length. This partitions the box \(B\) into sub-boxes as shown in Figure 12.6.1.
Let \(0=x_0 \lt x_1 \lt x_2=4\) be the endpoints of the subintervals of \([0,4]\) after partitioning. Draw a picture of Figure 12.6.1 by hand and label these endpoints on your drawing. Repeat this process with \(0=y_0 \lt y_1 \lt y_2 \lt y_3=6\) and \(0=z_0 \lt z_1 \lt z_2=8\text{.}\) What is the length \(\Delta x\) of each subinterval \([x_{i-1},x_i]\) for \(i\) from 1 to 2? the length of \(\Delta y\text{?}\) of \(\Delta z\text{?}\)
(b)
The partitions of the intervals \([0,4]\text{,}\) \([0,6]\) and \([0,8]\) partition the box \(B\) into sub-boxes. How many sub-boxes are there? What is volume \(\Delta V\) of each sub-box?
(c)
Let \(B_{ijk}\) denote the sub-box \([x_{i-1},x_i] \times [y_{j-1},y_j] \times [z_{k-1}, z_k]\text{.}\) Say that we choose a point \((x_{ijk}^*, y_{ijk}^*, z_{ijk}^*)\) in the \(i,j,k\)th sub-box for each possible combination of \(i,j,k\text{.}\) Write a couple of sentences to describe what physical quantity will \(\delta(x_{ijk}^*, y_{ijk}^*, z_{ijk}^*) \Delta V\) approximate and be sure to explain the meaning of \(\delta(x_{ijk}^*, y_{ijk}^*, z_{ijk}^*)\text{.}\)
(d)
We partitioned the space of inputs for \(\delta(x, y, z)\) into a few pieces. Write a few sentences about what you would change in the process above to improve the approximation of the total mass. Be sure to explain how you know that your changes will give a better approximation.
