Chapter 13.2, Problem 69E

To determine

To prove that first part of the Fundamental Theorem of Calculus for Vector-Valued Functions that if $r\left(t\right)$ is continuous on $\left[a,b\right]$, and $R\left(t\right)$ is antiderivative of $r\left(t\right)$, then ${\displaystyle {\int }_a^{b}r\left(t\right)dt=}R\left(b\right)-R\left(a\right)$ by using the Fundamental Theorem of Calculus I from single-variable calculus.