Question

If \(f(2)=2\hat i-\hat j+2\hat k\) and \(f(3)=4\hat i-2\hat j+3\hat k\), then the value of \(\int_2^3\left(f\cdot\dfrac{df}{dt}\right)dt\) is

• A. \(10\)
• B. \(11\)
• C. \(-10\)
• D. \(-11\)

Hint:

Use \(f\cdot f'= \frac{1}{2}\frac{d}{dt}(|f|^2)\) for integrals involving \(f\cdot \frac{df}{dt}\).

Answer 1

The Correct Option is A

Concept:
We use the identity \[ \frac{d}{dt}(f\cdot f)=2f\cdot \frac{df}{dt} \] Therefore, \[ f\cdot \frac{df}{dt} = \frac{1}{2}\frac{d}{dt}(f\cdot f) \]

Step 1: Apply the formula. \[ \int_2^3 f\cdot \frac{df}{dt}\,dt = \frac{1}{2}\int_2^3 \frac{d}{dt}(f\cdot f)\,dt \] \[ = \frac{1}{2}\left[|f(t)|^2\right]_2^3 \] \[ = \frac{1}{2}\left(|f(3)|^2-|f(2)|^2\right) \]

Step 2: Find \(|f(3)|^2\). \[ f(3)=4\hat i-2\hat j+3\hat k \] \[ |f(3)|^2=4^2+(-2)^2+3^2 \] \[ =16+4+9 \] \[ =29 \]

Step 3: Find \(|f(2)|^2\). \[ f(2)=2\hat i-\hat j+2\hat k \] \[ |f(2)|^2=2^2+(-1)^2+2^2 \] \[ =4+1+4 \] \[ =9 \]

Step 4: Substitute. \[ \int_2^3 f\cdot \frac{df}{dt}\,dt = \frac{1}{2}(29-9) \] \[ =\frac{20}{2} \] \[ =10 \]

Step 5: Final answer. \[ \boxed{10} \]