Question

Evaluate: \[ \int_0^4 |x - 1| \, dx \]

Hint:

Remember: To evaluate integrals involving absolute value, break the integral into intervals where the expression inside the absolute value is either positive or negative.

Answer 1

Step 1: Break the absolute value.
The absolute value function |x - 1| can be split into two parts based on the value of x. For x ≥ 1, |x - 1| = x - 1, and for x < 1, |x - 1| = 1 - x.

Therefore, we split the integral at x = 1:
∫₀¹ (1 − x) dx + ∫₁⁴ (x − 1) dx

Step 2: Integrate each part.
For the first integral:
∫₀¹ (1 − x) dx = [x − x²/2]₀¹ = 1 − 1/2 = 1/2

For the second integral:
∫₁⁴ (x − 1) dx = [x²/2 − x]₁⁴ = (16/2 − 4) − (1/2 − 1) = 8 − 4 − (−1/2) = 4 + 1/2 = 4.5

Step 3: Combine the results.
Total integral = 1/2 + 4.5 = 5