91Ó°ÊÓ

Graph the function and find its average value over the given interval. $$f(x)=3 x^{2}-3 \text { on }[0,1]$$

Find the areas of the regions enclosed by the lines and curves in Exercises \(63-72\). $$y=x^{2}-2 x \quad \text { and } \quad y=x$$

Archimedes \((287-212\) B.C.), inventor, military engineer, physicist, and the greatest mathematician of classical times in the Western world, discovered that the area under a parabolic arch is two-thirds the base times the height. Sketch the parabolic arch \(y=h-\left(4 h / b^{2}\right) x^{2}\) \(-b / 2 \leq x \leq b / 2,\) assuming that \(h\) and \(b\) are positive. Then use calculus to find the area of the region enclosed between the arch and the \(x\) -axis.

Find the area of the region in the first quadrant bounded by the line \(y=x,\) the line \(x=2,\) the curve \(y=1 / x^{2},\) and the \(x\) -axis.

Find the area of the region in the first quadrant bounded on the left by the \(y\) -axis, below by the line \(y=x / 4,\) above left by the curve \(y=1+\sqrt{x},\) and above right by the curve \(y=2 / \sqrt{x}\).

True, sometimes true, or never true? The area of the region between the graphs of the continuous functions \(y=f(x)\) and \(y=g(x)\) and the vertical lines \(x=a\) and \(x=b(a