Question

Use geometry to evaluate (see attached) (calculus) 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

Answer 1

check the picture below.

notice the area is just a 3x3 square and a triangle with a base of 3 and a height of 3, simply get the area of both, add them up, and that's the area of the piece-wise on that range.

Answer 2

13.5

Explanation:

The geometric representation of an integral is the area below the curve. So, let's graphic the function.

`f(x) = 3` is an horizontal line, (y = 3)

and
`f(x) = 6-x` is also a line, but it is decreasing.

We can see in the image that the area below the curve happens to be a polygon. In order to calculate the area we are going to separate it into simpler polygons: a square and a triangle.

The area of the square is
`3*3 = 9`.

The area of the triangle is
`(3*3)/(2) = (9)/(2) = 4.5`

If we add them, we get

`\int_0^6 f(x) dx =9 + 4.5 = 13.5`