Area of the Region Bounded by Curves

To find the area of the region bounded by the curves y = |x^2|, x = 1, x = 3, and the x-axis, we can break down the problem into smaller parts and calculate the areas separately.

1. Identifying the Bounded Region:
The given curves are y = |x^2|, x = 1, and x = 3.

- The curve y = |x^2| represents a "V" shape, opening upwards, with the vertex at the origin (0,0).
- The line x = 1 is a vertical line passing through the point (1,0).
- The line x = 3 is another vertical line passing through the point (3,0).

The bounded region is the area enclosed by these curves, the x-axis, and the vertical lines x = 1 and x = 3.

2. Calculating the Area:
To calculate the area of the bounded region, we need to integrate the function y = |x^2| with respect to x, from x = 1 to x = 3.

- Since the function y = |x^2| has a "V" shape and is symmetric about the y-axis, we can split the integral into two parts:
- The area above the x-axis, from x = 1 to x = 3.
- The area below the x-axis, from x = 1 to x = 3.

3. Area Above the x-axis:
To calculate the area above the x-axis, we need to integrate the function y = x^2 from x = 1 to x = 3.

- The integral of x^2 is (x^3)/3.
- Evaluating the integral from x = 1 to x = 3, we get:
- ((3^3)/3) - ((1^3)/3) = 9 - 1/3 = 8 2/3.

4. Area Below the x-axis:
To calculate the area below the x-axis, we need to integrate the function y = -x^2 from x = 1 to x = 3.

- The integral of -x^2 is -(x^3)/3.
- Evaluating the integral from x = 1 to x = 3, we get:
- -((3^3)/3) + ((1^3)/3) = -9 + 1/3 = -8 2/3.

5. Total Area of the Bounded Region:
To find the total area of the bounded region, we add the areas above and below the x-axis:

- Area above the x-axis: 8 2/3
- Area below the x-axis (considering the absolute value): 8 2/3

Adding both areas, we get:
8 2/3 + 8 2/3 = 17 1/3 = 1

Therefore, the correct answer is option 'B' - 1.