Question

\(\displaystyle \int_{0}^{\pi} \sin x \, dx = ?\)

• A. \(0\)
• B. \(1\)
• C. \(2\)
• D. \(-2\)

Hint:

For \(\int_{0}^{\pi} \sin x\,dx\), always remember the answer is \(2\), because the curve of \(\sin x\) remains positive between \(0\) and \(\pi\).

Answer 1

The Correct Option is C

Concept:
Definite integration is used to calculate the area under a curve between two given limits. Here, we have to integrate the trigonometric function \(\sin x\) from \(0\) to \(\pi\).

Step 1: Write the given integral. \[ I = \int_{0}^{\pi} \sin x \, dx \]

Step 2: Find the antiderivative of \(\sin x\). \[ \int \sin x \, dx = -\cos x \] Therefore, \[ I = \left[-\cos x\right]_{0}^{\pi} \]

Step 3: Apply the upper and lower limits. \[ I = -\cos \pi - \left(-\cos 0\right) \] We know that: \[ \cos \pi = -1 \] and \[ \cos 0 = 1 \]

Step 4: Substitute the values. \[ I = -(-1) - (-1) \] \[ I = 1 + 1 \] \[ I = 2 \] Therefore, \[ \int_{0}^{\pi} \sin x \, dx = 2 \] \[ \therefore \text{Correct Answer is (C)} \]