Answer
(a) See solution.
(b) See solution.
Work Step by Step
It is given that f(x) = C.
(a) We have to show geometrically that
\begin{align}
\int_{a}^{b}f(x) \ dx = C \times(b-a)
\end{align}
We know that if f(x) is constant, then the graph of the function is parallel to the x axis and equal to C. Also, we know that the definite integral calculates the area under the graph over the given interval. Thus, by considering those facts, it is possible to calculate the area under the graph f(x) over the interval [a, b] by using the formula:
\begin{align}
& Area = length \times width \\
& Area = \int_{a}^{b}f(x) \ dx = C \times(b-a)
\end{align}
(b) We have to use Definition 4.5.1 in order to show that
\begin{align}
\int_{a}^{b}f(x) \ dx = C \times(b-a)
\end{align}
From the Definition 4.5.1, we know that
\begin{align}
\lim\limits_{max \varDelta x_{k} \to 0} \sum_{k=1}^{n} f(x_{k}^{*})\varDelta x = \int_{a}^{b}f(x) \ dx
\end{align}
From the definition of the Riemann sum if f(x) = C:
\begin{align}
\lim\limits_{max \varDelta x_{k} \to 0} \sum_{k=1}^{n} f(x_{k}^{*})\varDelta x = C\times(b-a)
\end{align}
Thus,
\begin{align}
\int_{a}^{b}f(x) \ dx = C\times(b-a)
\end{align}
