Question

Find the average squared distance between the points of R={(x,y):0≤x≤2,0≤y≤4} and the point (2,4). The average squared distance is (Type an integer or a simplified fraction.)

Answer 1

To find the average squared distance between the points in set
`\(R = \{(x, y) : 0 \leq x \leq 2, 0 \leq y \leq 4\}\)` and the point (2,4), we set up the integral using the formula for average distance over a region, substitute the coordinates of the given point, calculate the area of
`\(R\)`, evaluate the integral over the region, and obtain the average squared distance as the final result.

Step-by-step explanation:

To find the average squared distance between the points in set
`\(R = \{(x, y) : 0 \leq x \leq 2, 0 \leq y \leq 4\}\)` and the point (2,4), follow these steps:

1. **Setup the Integral:**

The average squared distance can be represented as an integral over the region
`\(R\)` using the formula:

`\[ \text{Average Distance} = \frac{1}{\text{Area of } R} \int\int_R [(x_2 - x_1)^2 + (y_2 - y_1)^2] \,dx \,dy\]`

2. **Substitute the Point (2,4):**

Substitute the coordinates of the given point (2,4) into the integral:

`\[ \frac{1}{\text{Area of } R} \int\int_R [(x - 2)^2 + (y - 4)^2] \,dx \,dy\]`

3. **Calculate the Area of
`\(R\)`:**

The area of
`\(R\)` is given by the product of the ranges of x and y:

`\[ \text{Area of } R = (2-0) \cdot (4-0) = 8\]`

4. **Evaluate the Integral:**

Plug in the values into the integral:

`\[ (1)/(8) \int\int_R [(x - 2)^2 + (y - 4)^2] \,dx \,dy\]`

5. **Integrate Over the Given Region:**

Integrate over the region
`\(R\)` by calculating:

`\[ \int\int_R [(x - 2)^2 + (y - 4)^2] \,dx \,dy = \int_0^2 \int_0^4 [(x - 2)^2 + (y - 4)^2] \,dx \,dy\]`

6. **Perform the Integration:**

Integrate with respect to x and y separately:

`\[ \int_0^2 \left[\int_0^4 [(x - 2)^2 + (y - 4)^2] \,dy\right] \,dx\]`

7. **Calculate the Average Squared Distance:**

Finally, multiply the result by
`\((1)/(8)\)` to obtain the average squared distance.

The numerical result obtained from this calculation will be the answer to the problem, and it can be expressed as an integer or a simplified fraction.