91Ó°ÊÓ

Evaluate the definite integral. Use the integration capabilities of a graphing utility to verify your result. $$ \int_{1}^{2} \frac{x-2}{x} d x $$

Sketch the graph of the hypocycloid of four cusps \(x^{2 / 3}+y^{2 / 3}=4\) and find its perimeter.

Find the arc length of the graph of \(y=\sqrt{16-x^{2}}\) over the interval \([0,4]\).

Show that the length of one arch of the sine curve is equal to the length of one arch of the cosine curve.

(a) show that the nonnegative function is a probability density function, (b) find \(P(0 \leq x \leq 4)\), and(c) find \(E(x)\). $$ f(t)=\left\\{\begin{array}{ll} \frac{1}{7} e^{-t / 7}, & t \geq 0 \\ 0, & t<0 \end{array}\right. $$

Building Design The cross section of a precast concrete beam for a building is bounded by the graphs of the equations \(x=\frac{2}{\sqrt{1+y^{2}}}, x=\frac{-2}{\sqrt{1+y^{2}}}, y=0\), and \(y=3\) where \(x\) and \(y\) are measured in feet. The length of the beam is 20 feet (see figure). (a) Find the volume \(V\) and the weight \(W\) of the beam. Assume the concrete weighs 148 pounds per cubic foot. (b) Then find the centroid of a cross section of the beam.

Determine the constants \(a\) and \(b\) such that \(\sin x+\cos x=a \sin (x+b)\) Use this result to integrate \(\int \frac{d x}{\sin x+\cos x}\).

Apply the Extended Mean Value Theorem to the functions \(f\) and \(g\) on the given interval. Find all values \(c\) in the interval \((a, b)\) such that $$\frac{f^{\prime}(c)}{g^{\prime}(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}$$ Functions \(\quad\) Interval $$ f(x)=\frac{1}{x}, \quad g(x)=x^{2}-4 $$ $$ [1,2] $$

For the region bounded by the graphs of the equations, find (a) the volume of the solid formed by revolving the region about the \(x\) -axis and (b) the centroid of the region. \(y=\cos x, y=0, x=0, x=\pi / 2\)

Find the values of \(a\) and \(b\) such that \(\lim _{x \rightarrow 0} \frac{a-\cos b x}{x^{2}}=2\).