Question

The function f (x) is defined as `f (x) = {(-1, x ≤ 0),( (x + 2) ^ 2, 0 < x ≤ 3),(x, x > 3):}`. The area under the f (x) curve between x = (-2) and x = 5 is? Options: a) 27 b) 205/3 c) 155/3 d) 80

Is it to be solved using application of integral? Please help.

Answer 1

The answer is none of the above.

Explanation:

This function should be relatively easy to graph, so I would give that a try.

So pretty much, we have a rectangle with height `-1` and width `2`, the area under the parabola `(x +2)^2` on `[0, 3]` and a trapeze with `b_1 = 5`, `b_2 = 3`, `h = 2`.

So our expression for area will be

`A = (2)(-1) + int_0^3 x^2 +4x + 4dx + ((5 + 3)(2))/2`

`A = 6 + int_0^3 x^2 + 4x + 4 dx`

`A = 6 + [1/3x^3 + 2x^2 + 4x]_0^3`

`A = 6 + 1/3(3)^3 + 2(3)^2 + 4(3)`

`A = 6 + 9 + 18 + 12`

`A = 45`

Hopefully this helps!