Answer
(a)
\begin{align}
\int_{0}^{1}f(x)dx=1
\end{align}
(b)
\begin{align}
\int_{-1}^{1}f(x)dx=0
\end{align}
(c)
\begin{align}
\int_{1}^{10}f(x)dx=18
\end{align}
(d)
\begin{align}
\int_{\frac{1}{2}}^{5}f(x)dx=\frac{35}{4}
\end{align}
Work Step by Step
The given piecewise function is
\begin{align}
f(x) = \begin{cases}
2x, &\quad x\leq1 \\
2, &\quad x\gt1 \\
\end{cases}
\end{align}
(a)
\begin{align}
\int_{0}^{1}f(x)dx=\int_{0}^{1}2xdx=\big[ x^{2}\big]_{0}^{1} = 1
\end{align}
(b)
\begin{align}
\int_{-1}^{1}f(x)dx=\int_{-1}^{1}2xdx=\big[ x^{2}\big]_{-1}^{1} = 1 - 1=0
\end{align}
(c)
\begin{align}
\int_{1}^{10}f(x)dx=\int_{1}^{10}2dx=\big[2x\big]_{1}^{10} = 20-2 = 18
\end{align}
(d)
\begin{align}
\int_{\frac{1}{2}}^{5}f(x)dx=\int_{\frac{1}{2}}^{1}2xdx + \int_{1}^{5}2dx = \big[ x^{2}\big]_{\frac{1}{2}}^{1} + \big[2x\big]_{1}^{5} = 1 - \frac{1}{4} + 10 - 2 = \frac{35}{4}
\end{align}
