Problem 34 - Calculus Early Transcendentals

Question

Average value Compute the average value of the following functions over the region \(R\). $$f(x, y)=\sin x \sin y ; R=\\{(x, y): 0 \leq x \leq \pi, 0 \leq y \leq \pi\\}$$

Step-by-Step Solution

Answer

Question: Determine the average value of the function \(f(x, y) = \sin x \sin y\) over the region R, where R is the rectangle bounded by \(0 \leq x \leq \pi\) and \(0 \leq y \leq \pi\). Answer: The average value of the function \(f(x, y) = \sin x \sin y\) over the region R is \(\frac{4}{\pi^2}\).

Step 1: Write the formula for the average value

The formula for the average value of a function of two variables over a rectangular region is given by: $$\text{average value} = \frac{\iint_R f(x,y) \, dA}{\text{Area of R}}$$ where \(f(x, y)\) is the given function, \(R\) is the rectangular region, and \(dA=dxdy\).

Step 2: Calculate the area of the region R

The region \(R\) is defined by \(0 \leq x \leq \pi\) and \(0 \leq y \leq \pi\). The area of this rectangle is given by: $$ \text{Area of R} = (\text{length})\times(\text{width}) = (\pi - 0)\times(\pi - 0) = \pi^2 $$

Step 3: Set up the double integral to compute the average value

Using the formula for the average value and substituting in our function, we get: $$\text{average value} = \frac{1}{\pi^2} \iint_R \sin x \sin y \, dA$$ Since the region is rectangular, we can write the double integral as follows: $$\text{average value} = \frac{1}{\pi^2} \int_{0}^{\pi} \int_{0}^{\pi} \sin x \sin y \, dxdy$$

Step 4: Evaluate the double integral

Now, let's evaluate the double integral: 1. First, we evaluate the inner integral with respect to \(x\): $$\int_{0}^{\pi} \sin x \, dx = [-\cos x]_{0}^{\pi} = -\cos \pi - (-\cos 0) = -(-1) - (-1) = 2$$ 2. Next, we evaluate the outer integral with respect to \(y\): $$\int_{0}^{\pi} 2 \sin y \, dy = [2(-\cos y)]_{0}^{\pi} = 2(-\cos \pi - (-\cos 0)) = 4$$ 3. Now, we substitute the result back into the formula for the average value: $$\text{average value} = \frac{1}{\pi^2} \cdot 4$$

Step 5: Calculate the final result

Now, we can compute the average value: $$\text{average value} = \frac{4}{\pi^2}$$ So, the average value of the given function \(f(x, y)=\sin x \sin y\) over the region \(R\) is \(\frac{4}{\pi^2}\).