Which of the following statements are true?
(i) If \(\mathrm{C}>0\) is a constant, the region under the line
\(\mathrm{y}=\mathrm{C}\) on the interval \([\mathrm{a}, \mathrm{b}]\) has area
\(\mathrm{A}=\mathrm{C}(\mathrm{b}-\mathrm{a})\)
(ii) IfC> 0 is a constant and \(\mathrm{b}>\mathrm{a} \geq 0\), the region under
the line \(\mathrm{y}=\mathrm{Cx}\) on the interval \([\mathrm{a}, \mathrm{b}]\)
has
area \(\mathrm{A}=\frac{1}{2} \mathrm{C}(\mathrm{b}-\mathrm{a})\)
(iii) Theregion under the parabolay \(=\mathrm{x}^{2}\) on the interval
\([\mathrm{a}, \mathrm{b}]\) has area less than
\(\frac{1}{2}\left(\mathrm{~b}^{2}+\mathrm{a}^{2}\right)(\mathrm{b}-\mathrm{a})\).
(iv) The region under the curve \(\mathrm{y}=\sqrt{1-\mathrm{x}^{2}}\) on the
interval \([-1,1]\) has area \(\mathrm{A}=\frac{\pi}{2}\).
(v) Let f be a function that satisfies \(\mathrm{f}(\mathrm{x}) \geq 0\) for
\(\mathrm{x}\) in the interval \([a, b]\). Then the area under the curve
\(\mathrm{y}=\mathrm{f}^{2}(\mathrm{x})\) on the interval \([\mathrm{a},
\mathrm{b}]\) mustalways be greater than the area under
\(\mathrm{y}=\mathrm{f}(\mathrm{x})\) on the same interval.
(vi) If f is even and \(\mathrm{f}(\mathrm{x}) \geq 0\) throughout the interval
\([-a, a]\), then the area under the curve \(y=f(x)\) on this interval is twice
the area under \(\mathrm{y}=\mathrm{f}(\mathrm{x})\) on \([0, a]\).
