Use geometry to evaluate the integral from 0 to 6 of the function f of x, dx for f of x equals 3 for x less than or equal to 3 and equals the quantity 6 minus x for x greater than 3

- 27
- 13.5
-12
- 10.5

To solve this integral using geometry, we need to split the interval [0, 6] into two parts based on the given function f(x).

For x less than or equal to 3, the function f(x) is equal to 3. This means that the graph of f(x) is a horizontal line at y = 3 in this interval.

For x greater than 3, the function f(x) is equal to 6 - x. This means that the graph of f(x) is a line in this interval with a slope of -1 and y-intercept of 6.

To evaluate the integral, we need to find the area between the graph of f(x) and the x-axis in each interval, and then sum these areas together.

1) For the interval [0, 3]:
The graph of f(x) is a horizontal line at y = 3.
The area between the graph and the x-axis in this interval is a rectangle with a height of 3 and a base of 3. So the area is 3 * 3 = 9.

2) For the interval (3, 6]:
The graph of f(x) is a line with a slope of -1 and a y-intercept of 6.
To find the area between the graph and the x-axis in this interval, we need to find the area of a triangle with a base of 3 (since x goes from 3 to 6) and a height of (6 - 3) = 3. So the area is (1/2) * 3 * 3 = 4.5.

Now, we can sum the areas from both intervals:
Area = 9 + 4.5 = 13.5

Therefore, the value of the integral from 0 to 6 of the function f(x) is 13.5.

Hence, the correct answer is - 13.5 (since the function f(x) is decreasing in the interval [0, 6]).

To evaluate the given integral using geometry, we can break it down into two separate integrals and find the areas of the corresponding geometric shapes.

First, let's consider the interval from 0 to 3. In this interval, the function f(x) is equal to 3. Therefore, we have a rectangle with a base of 3 (since 3 - 0 = 3) and a height of 3. The area of this rectangle is given by base × height, which is 3 × 3 = 9.

Next, let's consider the interval from 3 to 6. In this interval, the function f(x) is equal to 6 - x. This represents a line segment with a negative slope. The area under this line segment can be found by calculating the area of a triangle.

The base of this triangle is given by 6 - 3 = 3 (since 6 - 3 = 3). The height of the triangle can be found by evaluating the function at x = 3, which gives us f(3) = 6 - 3 = 3. Therefore, the height of the triangle is 3.

Now, let's calculate the area of this triangle. The area of a triangle is given by base × height divided by 2. In this case, it is 3 × 3 / 2 = 9 / 2 = 4.5.

Finally, we add up the areas of the rectangle and the triangle to find the total area under the function f(x) from 0 to 6. 9 + 4.5 = 13.5.

Therefore, the value of the integral from 0 to 6 of the function f(x), dx is 13.5.

Therefore, the correct answer is - 13.5.

graph it!

You know how to find the area of rectangles and triangles.