Definite Integrals

For $\boldsymbol{r}(t) = \langle x(t), y(t), z(t) \rangle$ continuous on $[a, b]$, the definite integral over $[a, b]$ is defined componentwise:

For $\boldsymbol{r}(t)$, $\boldsymbol{r}_1(t)$, $\boldsymbol{r}_2(t)$ continuous on $[a, b]$ and scalar $k$:

• $\displaystyle\int_a^b k\boldsymbol{r}(t) \, \text{d}t = k \int_a^b \boldsymbol{r}(t) \, \text{d}t$
• $\displaystyle\int_a^b [\boldsymbol{r}_1(t) + \boldsymbol{r}_2(t)] \, \text{d}t = \int_a^b \boldsymbol{r}_1(t) \, \text{d}t + \int_a^b \boldsymbol{r}_2(t) \, \text{d}t$