Calculus Integration

Question

Find the position function  if , , and .

Answer 1

v(t) = ∫(6t - 2)dt = 3t² - 2t + C; v(0) = C = 3; so, v(t) = 3t² - 2t + 3.

x(t) = ∫v(t)dt = ∫(3t² - 2t + 3))dt = t³ - t² + 3t + D, x(0) = D = 7;

so, x(t) = t³ - 3t² + 3t + 7

Answer 2

Velocity v(t) is the indefinite integral of acceleration a(t).

v(t) = 3t² - 2t + C

The constant C is obtained from the constraint v(0) = 3, so C = 3.

v(t) = 3t² - 2t + 3

Position, x(t), is the indefinite integral of velocity v(t).

x(t) = t³ - t² + 3t + D

The constant D is obtained from the constraint x(0) = 7, so D = 7.

x(t) = t³ - t² + 3t + 7