91Ó°ÊÓ

Determine whether the statement is true or false. Explain your answer. [In each exercise, assume that \(f\) and \(g\) are distinct continuous functions on \([a, b]\) and that \(A\) denotes the area of the region bounded by the graphs of \(y=f(x)\), \(y=g(x), x=a\), and \(x=b .]\) If $$ A=\left|\int_{a}^{b}[f(x)-g(x)] d x\right| $$ then the graphs of \(y=f(x)\) and \(y=g(x)\) don't cross on \([a, b] .\)

True-False Determine whether the statement is true or false. Explain your answer. [In these exercises, assume that a solid \(S\) of volume \(V\) is bounded by two parallel planes perpendicular to the \(x\) -axis at \(x=a\) and \(x=b\) and that for each \(x\) in \([a, b], A(x)\) denotes the cross-sectional area of \(S\) perpendicular to the \(x\) -axis.] $$ \begin{aligned} &\text { The average value of } A(x) \text { on the interval }[a, b] \text { is given by }\\\ &V /(b-a) . \end{aligned} $$

Let \(y=f(x)\) be a smooth curve on \([a, b]\) and assume that \(f(x) \geq 0\) for \(a \leq x \leq b\). Let \(A\) be the area under the curve \(y=f(x)\) between \(x=a\) and \(x=b\), and let \(S\) be the area of the surface obtained when this section of curve is revolved about the \(x\) -axis. (a) Prove that \(2 \pi A \leq S\). (b) For what functions \(f\) is \(2 \pi A=S\) ?

Evaluate the integrals. $$ \int \cosh (2 x-3) d x $$

Use cylindrical shells to find the volume of the solid that is generated when the region that is enclosed by \(y=x^{3}\), \(y=1, x=0\) is revolved about the line \(y=1\).

Use a CAS to find the mass and center of gravity of the lamina with density \(\delta .\) A lamina bounded by the graphs of \(y=\cos x, y=\sin x\) \(x=0\), and \(x=\pi / 4 ; \delta=1+\sqrt{2}\).

Determine whether the statement is true or false. Explain your answer. [In Exercise 34 , assume that the (rotated) square lies in the \(x y\) -plane to the right of the \(y\) -axis.] The centroid of a rectangle is the intersection of the diagonals of the rectangle.

Evaluate the integrals. $$ \int \sqrt{\tanh x} \operatorname{sech}^{2} x d x $$

How might you recognize that a problem can be solved by means of the work- energy relationship? That is, what sort of "givens" and "unknowns" would suggest such a solution? Discuss two or three examples.

Estimate the value of \(k(0