Tích Phân Các Hàm Giá Trị Vector

Integrating a vector, one component at a time

• Differentiating a vector was component-wise. Integrating one is exactly the same.
• Integrate each component separately, and bundle the results back into a vector.
• $\displaystyle\int \langle f(t),\,g(t)\rangle\,dt = \Big\langle \int f\,dt,\ \int g\,dt\Big\rangle$.
• This lets you run motion backward: from acceleration to velocity, velocity to position.

Component-wise antiderivatives

• Each component gets its own antiderivative — and its own constant.
• The constants combine into a constant vector $\mathbf{C}=\langle C_1,\,C_2\rangle$.
• So the indefinite integral of a vector function is itself a vector (plus $\mathbf{C}$).
• Nothing new to learn — just apply single-variable integration twice.

The position vector points to the particle; the velocity vector is tangent to its path

Velocity from acceleration

• Given the acceleration vector $\mathbf{a}(t)$, integrate to recover velocity:
• $\mathbf{v}(t)=\displaystyle\int \mathbf{a}(t)\,dt$, with the constant vector fixed by an initial velocity $\mathbf{v}(t_0)$.
• Integrate again for position, using an initial position.
• It's the vector version of "position ← velocity ← acceleration" by integration.

Use the initial condition to find $\mathbf{C}$

• Each integration introduces a constant vector you must pin down.
• Plug in the given initial vector (velocity or position at a time) and solve for $\mathbf{C}$.
• Do this per component: the initial condition fixes both $C_1$ and $C_2$.
• Then write the fully determined vector function.