91Ó°ÊÓ

Use the most efficient strategy for computing the area of the following regions. The region in the first quadrant bounded by \(y=x^{-1}, y=4 x\), and \(y=x / 4\)

Find the volume of the solid generated in the following situations. The region \(R\) in the first quadrant bounded by the graphs of \(y=x\) and \(y=1+\frac{x}{2}\) is revolved about the line \(y=3\).

Two bars of length \(L\) have densities \(\rho_{1}(x)=4 e^{-x}\) and \(\rho_{2}(x)=6 e^{-2 x},\) for \(0 \leq x \leq L\) a. For what values of \(L\) is bar 1 heavier than bar \(2 ?\) b. As the lengths of the bars increase, do their masses increase without bound? Explain.

Let \(f(x)=x^{p}\) and \(g(x)=x^{1 / q},\) where \(p>1\) and \(q>1\) are positive integers. Let \(R_{1}\) be the region in the first quadrant between \(y=f(x)\) and \(y=x\) and let \(R_{2}\) be the region in the first quadrant between \(y=g(x)\) and \(y=x\) a. Find the area of \(R_{1}\) and \(R_{2}\) when \(p=q,\) and determine which region has the greater area. b. Find the area of \(R_{1}\) and \(R_{2}\) when \(p>q,\) and determine which region has the greater area. c. Find the area of \(R_{1}\) and \(R_{2}\) when \(p

Hooke's law is applicable to idealized (linear) springs that are not stretched or compressed too far. Consider a nonlinear spring whose restoring force is given by \(F(x)=16 x-0.1 x^{3},\) for \(|x| \leq 7\) a. Graph the restoring force and interpret it. b. How much work is done in stretching the spring from its equilibrium position \((x=0)\) to \(x=1.5 ?\) c. How much work is done in compressing the spring from its equilibrium position \((x=0)\) to \(x=-2 ?\)

Find the derivatives of the following functions. $$f(x)=x \sinh ^{-1} x-\sqrt{x^{2}+1}$$

Compute the following derivatives using the method of your choice. \(\frac{d}{d x}\left(e^{-10 x^{2}}\right)\)

Find the volume of the solid generated in the following situations. The region \(R\) is bounded by the graph of \(f(x)=2 x(2-x)\) and the \(x\) -axis. Which is greater, the volume of the solid generated when \(R\) is revolved about the line \(y=2\) or the volume of the solid generated when \(R\) is revolved about the line \(y=0 ?\) Use integration to justify your answer.

Consider the following velocity functions. In each case, complete the sentence: The same distance could have been traveled over the given time period at a constant velocity of _____. $$v(t)=2 t+6, \text { for } 0 \leq t \leq 8$$

Compute the following derivatives using the method of your choice. \(\frac{d}{d x}\left(x^{\tan x}\right)\)