Question

Consider the Assertion (A) and Reason (R) given below:
Assertion (A): \(\displaystyle\int_0^t \sin x\, dx = 1 - \cos t\)
Reason (R): \(\sin x\) is continuous in any closed interval \([0, t]\).

• A. Both A and R are true and R is the correct explanation of A
• B. Both A and R are true and R is not the correct explanation of A
• C. A is true but R is false
• D. A is false but R is true

Hint:

Evaluate the integral using the antiderivative of \(\sin x\) and check whether continuity is what allows the integral to be evaluated via the Fundamental Theorem of Calculus.

Answer 1

The Correct Option is A

Step 1: Evaluate the integral in the assertion directly. Since \(\int \sin x\, dx = -\cos x + C\), we get \(\int_0^t \sin x\, dx = [-\cos x]_0^t = -\cos t - (-\cos 0) = -\cos t + 1 = 1 - \cos t\). This matches the assertion exactly, so A is true.

Step 2: Check the reason. The sine function is continuous everywhere on the real line, so in particular it is continuous on any closed interval \([0,t]\). So R is also true.

Step 3: Check whether R explains A. A continuous function on a closed bounded interval is Riemann integrable there, so the continuity of \(\sin x\) on \([0,t]\) is precisely what guarantees the definite integral in A exists and can be evaluated using the Fundamental Theorem of Calculus.

Step 4: Therefore both A and R are true, and R is indeed the correct explanation of A. The answer is option (A).

\[\boxed{\text{Both A and R are true and R is the correct explanation of A}}\]