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Calculate the total area of the region described. Do not count area beneath the x-axis as negative. The region is bounded by the line y = x, the x-axis, and the lines x = 0 and x = 3.

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Final answer:

The total area of the region is 4.5 square units, calculated by adding the area of the triangle to the area of the rectangle.

Step-by-step explanation:

To calculate the total area of the region described, we need to calculate the area under the graph bounded by the line y = x, the x-axis, and the lines x = 0 and x = 3. This region forms a triangle and a rectangle. The area of the triangle can be calculated using the formula: A = 0.5 * base * height, where the base is the distance between x = 0 and x = 3 (which is 3 - 0 = 3) and the height is the difference between the y-values of the line y = x at x = 0 and x = 3 (which is 0 - 3 = -3). The area of the triangle is 0.5 * 3 * -3 = -4.5 square units. The area of the rectangle can be calculated by multiplying its length, which is 3 units, by its height, which is 3 units (since the line y = x intersects the x-axis at (0, 0) and (3, 3)). The area of the rectangle is 3 * 3 = 9 square units. Therefore, the total area of the region is -4.5 + 9 = 4.5 square units.

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