Question

The value of the line integral ∫ from P to Q [ z² dx + 3y² dy + 2xz dz ] along the straight line joining the points P(1,1,2) and Q(2,3,1) is _____________

• A. 20
• B. 24
• C. 29
• D. $-5$

Hint:

For line integrals along a straight segment, parametrize with $t\in[0,1]$ using $r(t)=P+t(Q-P)$, compute $dx,dy,dz$, substitute, and integrate.

Answer 1

The Correct Option is B

Parametrize the straight line PQ.
Let t ∈ [0,1], with x = 1 + t, y = 1 + 2t, z = 2 − t.
Then dx = dt, dy = 2 dt, dz = −dt.

Substitute into the integrand.
z² dx = (2 − t)² dt,
3y² dy = 3(1 + 2t)² (2 dt) = 6(1 + 2t)² dt,
2xz dz = 2(1 + t)(2 − t)(−dt) = −2(1 + t)(2 − t) dt.

Integrate from t = 0 to t = 1.
∫₀¹ [ (2 − t)² + 6(1 + 2t)² − 2(1 + t)(2 − t) ] dt = ∫₀¹ (27t² + 18t + 6) dt = [ 9t³ + 9t² + 6t ]₀¹ = 9 + 9 + 6 = 24.

Final Answer:
24