Question

The following table gives the density (in units of \(\mathrm{g} / \mathrm{cm}^{2}\) ) at selected points of a thin semicircular plate of radius 3. Estimate the mass of the plate and explain your method. $$\begin{array}{|c|c|c|c|c|c|} \hline & \boldsymbol{\theta}=\mathbf{0} & \boldsymbol{\theta}=\boldsymbol{\pi} / \boldsymbol{4} & \boldsymbol{\theta}=\boldsymbol{\pi} / \boldsymbol{2} & \boldsymbol{\theta}=\boldsymbol{3} \pi / \boldsymbol{4} & \boldsymbol{\theta}=\boldsymbol{\pi} \\ \hline \boldsymbol{r}=\mathbf{1} & 2.0 & 2.1 & 2.2 & 2.3 & 2.4 \\ \hline \boldsymbol{r}=\mathbf{2} & 2.5 & 2.7 & 2.9 & 3.1 & 3.3 \\ \hline \boldsymbol{r}=\mathbf{3} & 3.2 & 3.4 & 3.5 & 3.6 & 3.7 \\ \hline \end{array}$$

Short answer

Question: Based on the given densities of a semicircular plate in a table using polar coordinates, estimate the mass of the semicircular plate using the trapezoidal rule for double integration. Answer: The estimated mass of the semicircular plate is approximately 24.22 g.

Step-by-step solution

Convert polar coordinates to Cartesian coordinates

The table uses polar coordinates, so we should convert them to Cartesian coordinates to make it easier to understand and visualize the problem. For each polar coordinate (r, θ), we can use the following conversion formulas: $$x = r\cos\theta$$ $$y = r\sin\theta$$ The plate lies on the xy-plane where the origin is the center of the circle and the semicircle is formed by connecting points (0,0) to (3,0).

Approximate mass calculation using trapezoidal rule

First, we should divide the semicircle into smaller regions based on the given points in the table. We'll get areas in the form of a grid. Next, let's use the trapezoidal rule to approximate the double integral which will estimate the mass. The trapezoidal rule formula for double integral is: $$\iint\limits_{A} \rho(x, y) \,dA \approx \frac{\Delta x \Delta y}{4} \sum_{i=0}^{n-1}\sum_{j=0}^{m-1}(\rho_{i,j} + \rho_{i+1,j}+\rho_{i,j+1}+\rho_{i+1,j+1})$$ where \(\rho_{i,j}\) is the density at point (i, j), \(\Delta x\) and \(\Delta y\) are the widths of the rectangles formed in x and y-direction, and n and m are the number of sub-intervals considered in x and y-direction respectively. In our case, for an interval from the angles θ to θ + \(\Delta\theta\) and the radius r to r + \(\Delta r\) in polar coordinates, the area in Cartesian coordinates can be estimated by: $$\Delta A = r\Delta r \Delta\theta$$ So, the formula becomes: $$M\approx \sum_{r=1}^3\sum_{\theta=0}^{\pi} \frac{\rho(r,\theta)r\Delta r\Delta\theta}{4}$$

Calculate the mass

We'll now use the data given in the table to plug into the formula derived in step 2 and sum up the individual terms to approximate the mass of the semicircular plate. Using the provided densities and intervals, we can compute: $$M \approx \frac{(\pi-0)(3-1)}{4}[2.0 + 2.1 + 2.2 + 2.3 + 2.4 + 2.5 + 2.7 + 2.9 + 3.1 + 3.3 + 3.2 + 3.4 + 3.5 + 3.6 + 3.7].$$ $$M \approx \frac{\pi(2)(38.7)}{4}.$$ Approximating the mass yields: $$M\approx 24.22\,\text{g}$$ Therefore, the estimated mass of the semicircular plate is about 24.22 g.

Key concepts

Double Integral

In calculus, a double integral is an extension of the concept of a single integral to functions of two variables, like the density function in our exercise. This type of integral helps to find quantities like area, volume, or in our case, mass over a particular region. When dealing with a function defined over a two-dimensional region, the double integral allows for summing up function values across that region. This accumulation can be thought of as stacking infinite slabs together to cover the area.
For the semicircular plate, we're solving a double integral over it since we're calculating its mass from varying densities at different points. The process involves integrating the density \(\rho(x,y)\) over the area A:
\[\iint_{A} \rho(x,y) \, dA\]
Handled correctly, the double integral blends the variations in density and shapes an understanding of how the total mass arises from contributing factors distributed over a region.

Trapezoidal Rule

The trapezoidal rule is a numerical method used to approximate definite integrals, especially when dealing with complex shapes or when function values aren't easily integrable using analytical methods. Imagine estimating the area under a curve by dividing it into a series of trapezoids, calculating the area for each, and adding them up. This rule offers an efficient way to handle non-linear plots or varied data points, which is precisely what we encounter in density tables.
For double integrals, like those used in our problem, the trapezoidal rule helps estimate the mass by calculating the area and density for small slices over the semicircle. Each small slice is essentially treated like a trapezoid, with the length of the top and bottom determined by the function's value at the endpoints. By adding together all these small masses, we get a good estimation of the mass of the entire plate.

• Use slice boundaries to configure the trapezoids over the region of interest.
• Calculate each slice's contribution to the total using densities from the table.
  

Hence, the trapezoidal rule makes numerical integration accessible and practical for real-world applications where analytic solutions are tough.

Polar Coordinates

Polar coordinates provide a clever way to describe positions in a plane using a radius and an angle, as opposed to standard x and y dimensions. Particularly useful in circular systems, they make it easier to describe curves or circular objects like the semicircular plate.
The conversion formula finds the Cartesian coordinates, helping visualize the layout with more traditional grids. With polar coordinates:
\(x = r \cos(\theta)\)
\(y = r \sin(\theta)\)
This approach aids in handling problems involving symmetry about a point, such as semicircles, circles, or circular sectors.
In the given problem, we're using values \(\theta\) and \(r\) from the table to examine densities on the semicircle and transition calculations to areas in x and y domains for the trapezoidal approximation. Converting to Cartesian coordinates simplifies understanding of the plate's geometric layout while retaining the essential properties of our original problem.

Density Estimation

Density estimation refers to determining the mass or weight spread over a given area, using data points collected at specific coordinates. The data table from the problem gives densities at certain r (radius) and \(\theta\) (angle) positions in a semicircle, which will be used to calculate the plate's total mass.
Basically, to approximate the mass, we treat each density reading as applying uniformly over small areas defined by radii and angles, adding them together after scaling by the small area's size. The table derives the density function instances we use for numerical integration, providing values for the double integral calculation.
Critical factors include:

• Understanding each density reading's local impact.
• Accuracy in how the numerical method sums these slices.
• Techniques like the trapezoidal rule guide this sum's balance.

Thus, density estimation morphs a raw dataset into tangible insights about a material's weight and distribution, indispensable for engineering, physics, and beyond.