Use the rectangles with the given dimensions to approximate the area of the region in the first quadrant bounded by the graphs of y = 4 – x2 , y = 0, and x = 0.


4 _
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1 2

To approximate the area of the region, we can divide it into smaller rectangles and calculate the area of each rectangle.

First, let's look at the dimensions of the rectangles given in the graph. The width of each rectangle is 1 because the distance between each vertical line (x = 0, x = 1, x = 2) is equal to 1.

Now, let's determine the height of each rectangle. We can do this by calculating the difference between the y-values of the two curves at each x-value.

For the first rectangle, the height is given by the difference between the y-values of the curves y = 4 - x^2 and y = 0 at x = 0. Since y = 0 at x = 0, the height of the rectangle is given by y = 4 - x^2, evaluated at x = 0. So the height of the first rectangle is 4 - 0^2 = 4.

For the second rectangle, the height is given by the difference between the y-values of the curves y = 4 - x^2 and y = 0 at x = 1. So the height of the second rectangle is 4 - 1^2 = 3.

For the third rectangle, the height is given by the difference between the y-values of the curves y = 4 - x^2 and y = 0 at x = 2. So the height of the third rectangle is 4 - 2^2 = 0.

Now, let's calculate the area of each rectangle by multiplying the width and height:

Area of first rectangle = width * height = 1 * 4 = 4 square units

Area of second rectangle = width * height = 1 * 3 = 3 square units

Area of third rectangle = width * height = 1 * 0 = 0 square units

To approximate the total area of the region, we add up the areas of all the rectangles:

Total Area = Area of first rectangle + Area of second rectangle + Area of third rectangle
= 4 + 3 + 0
= 7 square units

Therefore, the approximate area of the region in the first quadrant bounded by the graphs of y = 4 - x^2, y = 0, and x = 0 is 7 square units.