Problem 4 - Calculus Volume 3

Question

In the following exercises, estimate the volume of the solid under the surface \(z=f(x, y)\) and above the rectangular region \(R\) by using a Riemann sum with \(m=n=2\) and the sample points to be the lower left corners of the subrectangles of the partition. $$ f(x, y)=\cos x+\cos y, \quad R=[0, \pi] \times\left[0, \frac{\pi}{2}\right] $$

Step-by-Step Solution

Answer

The estimated volume is \(\frac{\pi^2}{2} + \frac{\pi^2 \sqrt{2}}{8}\).

Step 1: Divide the Region into Subrectangles

The rectangular region \(R\) is given by \([0, \pi] \times [0, \frac{\pi}{2}]\). We are asked to use a Riemann sum with \(m=n=2\), meaning we will divide \(R\) into 2 equal parts along the \(x\)-axis and 2 equal parts along the \(y\)-axis. This results in 4 subrectangles, each with dimensions \(\frac{\pi}{2} \times \frac{\pi}{4}\).

Step 2: Identify Sample Points and Evaluate the Function

For each subrectangle, we take the sample point to be the lower left corner. The coordinates of these points are given by 1. \((0, 0)\) 2. \((0, \frac{\pi}{4})\) 3. \((\frac{\pi}{2}, 0)\) 4. \((\frac{\pi}{2}, \frac{\pi}{4})\)Evaluate the function \(f(x, y) = \cos x + \cos y\) at each sample point:- \(f(0, 0) = \cos 0 + \cos 0 = 2\)- \(f(0, \frac{\pi}{4}) = \cos 0 + \cos \frac{\pi}{4} = 1 + \frac{\sqrt{2}}{2}\)- \(f(\frac{\pi}{2}, 0) = \cos \frac{\pi}{2} + \cos 0 = 1\)- \(f(\frac{\pi}{2}, \frac{\pi}{4}) = \cos \frac{\pi}{2} + \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}\)

Step 3: Calculate the Riemann Sum

The area of each subrectangle is \(\Delta A = \frac{\pi}{2} \times \frac{\pi}{4} = \frac{\pi^2}{8}\). Multiply the value of the function at each sample point by this area and sum them:\[V \approx \frac{\pi^2}{8}\left(2 + \left(1 + \frac{\sqrt{2}}{2}\right) + 1 + \frac{\sqrt{2}}{2}\right) = \frac{\pi^2}{8} \left(4 + \sqrt{2}\right)\]

Step 4: Simplify the Expression

Simplify the expression for the Riemann sum:\[V \approx \frac{\pi^2}{8}(4 + \sqrt{2}) = \frac{\pi^2}{8} \times 4 + \frac{\pi^2}{8} \times \sqrt{2} = \frac{\pi^2}{2} + \frac{\pi^2 \sqrt{2}}{8}\]This is the estimated volume of the solid under the surface over the region \(R\).

Key Concepts

Surface Volume Estimation

Estimating the volume of a solid under a surface and above a specific region is a fascinating application of calculus. This process involves summing up the volume of thin, rectangular slices, also known as subrectangles, over the area of interest. In the context of this exercise, we want to find the volume of the solid lying below a surface described by the function \( z = f(x, y) = \cos x + \cos y \) and above the rectangular region \( R \) defined by \( [0, \pi] \times [0, \frac{\pi}{2}] \).

A Riemann sum is a method used in calculus to estimate the total value, in this case volume, by approximating the surface with these small subrectangles. By calculating the height at specific points and multiplying by the area, we can gain an approximation of the volume. As we increase the number of subrectangles, our approximation becomes closer to the real volume.

Rectangular Region Subdivision

To estimate the volume, we start by subdividing the given rectangular region \( R \). Here, the region \([0, \pi] \times [0, \frac{\pi}{2}]\) needs to be divided into subrectangles. Using the parameters \( m = 2 \) and \( n = 2 \) means dividing the region into 2 sub-intervals along each axis. This results in 4 smaller rectangles, each of equal size, within the original region.

Specifically, each subrectangle is formed by dividing the intervals \([0, \pi]\) along the \(x\)-axis into two parts, creating an interval length of \(\frac{\pi}{2}\), and the interval \([0, \frac{\pi}{2}]\) along the \(y\)-axis into two parts, creating an interval length of \(\frac{\pi}{4}\). Each subrectangle has dimensions of \(\frac{\pi}{2} \times \frac{\pi}{4}\).

Function Evaluation at Sample Points

Once the rectangular region is divided into subrectangles, we need to evaluate the function at specific points called sample points. In this Riemann sum approach, we use the lower left corner of each subrectangle as the sample point. These points for the exercise are:

• \((0, 0)\)
• \((0, \frac{\pi}{4})\)
• \((\frac{\pi}{2}, 0)\)
• \((\frac{\pi}{2}, \frac{\pi}{4})\)

Evaluating the function \( f(x, y) = \cos x + \cos y \) at these points gives us the height of each subrectangle. It provides values that reflect the height of the surface at those sample points:

• At \((0, 0)\), \( f(0, 0) = 2 \)
• At \((0, \frac{\pi}{4})\), \( f(0, \frac{\pi}{4}) = 1 + \frac{\sqrt{2}}{2} \)
• At \((\frac{\pi}{2}, 0)\), \( f(\frac{\pi}{2}, 0) = 1 \)
• At \((\frac{\pi}{2}, \frac{\pi}{4})\), \( f(\frac{\pi}{2}, \frac{\pi}{4}) = \frac{\sqrt{2}}{2} \)

Simplification of Expressions

After evaluating the function at each sample point, the next step involves simplifying the resulting Riemann sum expressions. From the exercise, the area of each subrectangle is \( \Delta A = \frac{\pi^2}{8} \).

To find the approximate volume, multiply the height of each subrectangle, as given by the function evaluations, by this area. Then, sum these products:
\[ V \approx \frac{\pi^2}{8} \left( 2 + \left(1 + \frac{\sqrt{2}}{2}\right) + 1 + \frac{\sqrt{2}}{2} \right) = \frac{\pi^2}{8} \left( 4 + \sqrt{2} \right) \] Finally, simplify the expression as follows:
\[ V \approx \frac{\pi^2}{2} + \frac{\pi^2 \sqrt{2}}{8} \] This provides us with the estimated volume of the solid lying under the curve above the specified region.