Solution
= Solution
Parametrize the <curve> by $\mathbf r(x)=(x,e^x,e^x)$ for $0\le x\le1$. Then $d\mathbf r=(1,e^x,e^x)\,dx$ and $\mathbf F(\mathbf r(x))=(e^{2x},x,e^x)$. Their <dot product> gives the <line integral>
$$
\int_C\mathbf F\cdot d\mathbf r=\int_0^1\left(2e^{2x}+xe^x\right)dx.
$$
Using <integration by parts> for the second term,
$$
\int_C\mathbf F\cdot d\mathbf r
=\bigl[e^{2x}+(x-1)e^x\bigr]_0^1,
\qquad\boxed{\int_C\mathbf F\cdot d\mathbf r=e^2.}
$$