Integrating Vector-Valued Functions
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.
To integrate $\langle f(t),g(t)\rangle$, you...
Integrate componentwise, plus a constant vector.
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.
Integrating a vector function introduces a constant vector $\mathbf{C}$.
One constant per component, bundled into $\mathbf{C}$.
Each component of the integral has its own constant of integration.
They combine into the constant vector $\mathbf{C}$.
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.
Integrate acceleration to velocity
y = ax² + bx
Integrating each acceleration component (plus a constant) recovers velocity — the initial condition fixes the constants.
Integrating the acceleration vector recovers the...
Acceleration integrates to velocity.
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.
If $\mathbf{a}(t)=\langle 2,6t\rangle$ and $\mathbf{v}(0)=\langle 1,0\rangle$, then $\mathbf{v}(t)=$
Integrate then apply $\mathbf{v}(0)$: $C_1=1,C_2=0$.
To determine the constant vector, apply the given ____ condition.
Plug in the initial velocity or position.
Integrate each component separately, and don't forget a constant vector $\mathbf{C}$ — a constant of integration for each component. Use the initial condition (velocity or position at a given time) to solve for $\mathbf{C}$ component by component before reporting the answer.
If $\mathbf{a}(t)=\langle 2,\ 6t\rangle$ and $\mathbf{v}(0)=\langle 1,\ 0\rangle$, find $\mathbf{v}(t)$.
- Integrate: $\mathbf{v}(t)=\langle 2t+C_1,\ 3t^2+C_2\rangle$.
- Apply $\mathbf{v}(0)=\langle 1,0\rangle$: $C_1=1$, $C_2=0$.
- $\mathbf{v}(t)=\langle 2t+1,\ 3t^2\rangle$.
Integrate a vector-valued function component by component: $\int\langle f,g\rangle\,dt=\langle\int f\,dt,\int g\,dt\rangle$, plus a constant vector $\mathbf{C}$. Integrate acceleration to get velocity (and again for position), using an initial condition to solve for $\mathbf{C}$ in each component.