Question

Evaluate the integral. 5 −5 f(x) dx where f(x) = 5 if −5 ≤ x ≤ 0 25 − x² if 0 < x ≤ 5

Answer 1

The integral ∫5 -5 f(x) dx, where f(x) changes its expression at x = 0, is split into two parts and calculated separately for [-5, 0] and (0, 5]. The integral evaluates to 150 - ⅓×(125).

Step-by-step explanation:

Evaluating the Given Integral

To evaluate the integral of the function f(x) from -5 to 5, we need to split the integral into two parts, since the function has two different expressions over different intervals. For the interval [-5, 0], f(x) is a constant function, so the integral is simple. For the interval (0, 5], f(x) is defined as 25 - x², a quadratic function. Thus, our integral is evaluated as follows:

∫5 -5 f(x) dx = ∫0 -5 5 dx + ∫5 0 (25 - x²) dx
= [5x]|0 -5 + [25x - ⅓x³]|5 0
= (5×0 - 5×(-5)) + (25×5 - ⅓×5³) - (25×0 - ⅓×0³)
= 25 + 125 - (⅓×5³)
= 25 + 125 - ⅓×(125)
= 150 - ⅓×(125)
= 150 - ⅓×(125).