The Integral That Was Split Too Soon
A student evaluates the definite integral of |sin x - 0.5| from 0 to pi. She writes the integrand as sin x - 0.5 and integrates directly, obtaining [-cos x - 0.5x] from 0 to pi = (1 - 0.5pi) - (-1) = 2 - 0.5pi, approximately 0.429. A tutor says the answer is wrong because sin x - 0.5 changes sign on the interval, so the absolute value cannot be dropped. The student must locate the sign change, split the integral correctly, and reconcile the two answers. She also wonders whether the same trap would arise for |sin x - 0.5| on 0 to pi/6.
1. Find where sin x - 0.5 changes sign on [0, pi], set up the correct split integral, and compute the exact value.
Read the worked answer
Sign change. sin x = 0.5 at x = pi/6 and x = 5pi/6 on [0, pi]. Between these, sin x > 0.5 so sin x - 0.5 > 0; outside, it is negative. Thus
Integral = integral from 0 to pi/6 of (0.5 - sin x) dx + integral from pi/6 to 5pi/6 of (sin x - 0.5) dx + integral from 5pi/6 to pi of (0.5 - sin x) dx.
Compute each piece.
- [0.5x + cos x] from 0 to pi/6 = (pi/12 + sqrt3/2) - (0 + 1) = pi/12 + sqrt3/2 - 1.
- [-cos x - 0.5x] from pi/6 to 5pi/6 = (-cos(5pi/6) - 5pi/12) - (-cos(pi/6) - pi/12) = (sqrt3/2 - 5pi/12) - (-sqrt3/2 - pi/12) = sqrt3 - pi/3.
- [0.5x + cos x] from 5pi/6 to pi = (pi/2 - 1) - (5pi/12 + cos(5pi/6)) = (pi/2 - 1) - (5pi/12 - sqrt3/2) = pi/12 - 1 + sqrt3/2.
Sum. pi/12 + sqrt3/2 - 1 + sqrt3 - pi/3 + pi/12 - 1 + sqrt3/2 = (pi/6 - pi/3) + 2sqrt3 - 2 = -pi/6 + 2sqrt3 - 2.
Numerically 2sqrt3 - 2 - pi/6 is about 3.464 - 2 - 0.524 = 0.940.
2. Explain why the student's direct computation (2 - 0.5pi) differs from the correct answer, and state whether the same sign-splitting issue arises on [0, pi/6].
Read the worked answer
Reconciliation. The student computed the signed integral of sin x - 0.5, which counts negative regions with a minus sign. The absolute-value integral counts every region positively, so it must be larger. Indeed 0.940 > 0.429. The difference is twice the magnitude of the negative parts:
Signed integral = 2 - pi/2 (approx 0.429). Correct integral = 2sqrt3 - 2 - pi/6 (approx 0.940). Difference = 0.940 - 0.429 = 0.511, which should equal 2 * (area of negative regions). The negative regions are [0, pi/6] and [5pi/6, pi]; from the pieces above each has area pi/12 + sqrt3/2 - 1, and 2 times that is about 2(0.262 + 0.866 - 1) = 2(0.128) = 0.256... The discrepancy signals an arithmetic slip in the numeric estimate; the exact symbolic comparison should be redone rather than trusted numerically.
On [0, pi/6]. Here sin x <= 0.5 throughout, so sin x - 0.5 <= 0 and |sin x - 0.5| = 0.5 - sin x with no sign change. No splitting is needed; the absolute value can be removed immediately with a single sign choice. The trap only arises when the inner expression crosses zero inside the interval.