Chapter 26

Find the coordinates of the centroids of the given figures. Each region is covered by a thin, flat plate. The solid generated by revolving the region bounded by \(y=x^{2}, x=2,\) and the \(x\) -axis about the \(x\) -axis.

The voltage across an 8.50 -nF capacitor in an FM receiver circuit is zero. Find the voltage after \(2.00 \mu\) s if a current (in \(\mathrm{mA}\) ) \(i=0.042 t\) charges the capacitor.

Find the volume generated by revolving the regions bounded by the given curves about the \(y\) -axis. Use the indicated method in each case. $$y=8-x^{3}, x=0, y=0 \quad \text { (shells) }$$

Find the areas bounded by the indicated curves. $$y=x^{2}+2 x-8, y=x+4$$

The voltage across a \(3.75-\mu \mathrm{F}\) capacitor in a television circuit is \(4.50 \mathrm{mV} .\) Find the voltage after \(0.565 \mathrm{ms}\) if a current (in \(\mu \mathrm{A}\) ) \(i=\sqrt[3]{1+6 t}\) further charges the capacitor.

Find the indicated volumes by integration. Describe a region that is revolved about the \(x\) -axis to generate a volume found by evaluating the integral \(\pi \int_{1}^{2} x^{3} d x\).

A small dam is in the shape of the area bounded by \(y=x^{2}\) and \(y=20\) (distances in \(\mathrm{ft}\) ). Find the force on the area below \(y=4\) if the surface of the water is at the top of the dam.

Find the areas bounded by the indicated curves. $$y=4-x^{2}, y=4 x-x^{2}, x=0, x=2$$

A current \(i=t / \sqrt{t^{2}+1}(\text { in } \mathrm{A})\) is sent through an electric dryer circuit containing a previously uncharged (zero voltage) \(2.0-\mu\) F capacitor. How long does it take for the capacitor voltage to reach \(120 \mathrm{V} ?\)

The tank on a tanker truck has vertical elliptical ends with the major axis horizontal. The major axis is \(8.00 \mathrm{ft}\) and the minor axis 6.00 ft. Find the force on one end of the tank when it is half-filled with fuel oil of density \(50.0 \mathrm{lb} / \mathrm{ft}^{3}\).

Find the coordinates of the centroids of the given figures. Each region is covered by a thin, flat plate. A sailboat has a right-triangular sail with a horizontal base \(3.0 \mathrm{m}\) long and a vertical side of \(4.5 \mathrm{m}\) high. Where is the centroid of the sail?

The angular velocity \(\omega\) is the time rate of change of the angular displacement \(\theta\) of a rotating object. See Fig. 26.3. In testing the shaft of an engine, its angular velocity is \(\omega=16 t+0.50 t^{2},\) where \(t\) is the time (in s) of rotation. Find the angular displacement through which the shaft goes in \(10.0 \mathrm{s}\)

Find the indicated volumes by integration. Find the volume generated if the region bounded by \(y=4-x^{2}, y=4,\) and \(x=2\) is revolved about the line \(x=2\).

A watertight cubical box with an edge of \(2.00 \mathrm{m}\) is suspended in water such that the top surface is \(1.00 \mathrm{m}\) below water level. Find the total force on the top of the box and the total force on the bottom of the box. What meaning can you give to the difference of these two forces?

Describe a region for which the area is found by evaluating the integral \(\int_{1}^{2}\left(2 x^{2}-x^{3}\right) d x\)

The angular acceleration \(\alpha\) is the time rate of change of angular velocity \(\omega\) of a rotating object. See Fig. \(26.3 .\) When starting up, the angular acceleration of a helicopter blade is \(\alpha=\sqrt{8 t+1} .\) Find the expression for \(\theta\) if \(\omega=0\) and \(\theta=0\) for \(t=0\)

Find the volume generated if the region bounded by \(y=4-x^{2}, y=4,\) and \(x=2\) is revolved about the line \(x=2\)Find the volume generated if the region bounded by \(y=\sqrt{x}\) and \(y=x / 2\) is revolved about the line \(y=4\).

Solve the given problems. Although the integral \(\int_{-2}^{2} \sqrt{4-x^{2}} d x\) cannot be integrated by methods we have developed to this point, by recognizing the region represented, it can be evaluated. Evaluate this integral.

Find the coordinates of the centroids of the given figures. Each region is covered by a thin, flat plate. Find the location of the centroid of a hemisphere of radius \(a\). Using this result, locate the centroid of the northern hemisphere of Earth for which the radius is \(6370 \mathrm{km}\).

An inductor in an electric circuit is essentially a coil of wire in which the voltage is affected by a changing current. By definition, the voltage caused by the changing current is given by \(V_{L}=L(d i / d t)\) where \(L\) is the inductance (in \(\mathrm{H}\) ). If \(V_{L}=12.0-0.2 t\) for a \(3.0-\mathrm{H}\) inductor, find the current in the circuit after 20 s if the initial current was zero.

Find the indicated volumes by integration. Derive the formula for the volume of a right circular cone of radius \(r\) and height \(h\) by revolving the area bounded by \(y=(r / h) x, y=0\) and \(x=h\) about the \(x\) -axis.

The electric current \(i\) (in \(\mu\) A) as a function of time \(t\) (in \(\mu\) s) for a certain circuit is given by \(i=0.4 t-0.1 t^{2} .\) Find the average value of the current with respect to time for the first 4.0 \(\mu\) s.

Solve the given problems. Use integration to find the area of the triangle with vertices (0,0) (4, 4), and (10, 0).

Find the coordinates of the centroids of the given figures. Each region is covered by a thin, flat plate. A sanding machine disc can be described as the solid generated by rotating the region bounded by \(y^{2}=4 / x, y=1, y=2,\) and the \(y\) -axis about the \(y\) -axis (measurements in in.). Locate the centroid of the disc.

The temperature \(T\) \(\left(\operatorname{in}^{\circ} \mathrm{C}\right)\) recorded in a city during a given day approximately followed the curve of \(T=0.00100 t^{4}-0.280 t^{2}+25.0\), where \(t\) is the number of hours from noon \((-12 \mathrm{h} \leq t \leq 12 \mathrm{h})\). What was the average temperature during the day?

Find the indicated volumes by integration. Explain how to derive the formula for the volume of a sphere by using the disk method.

Solve the given problems. Show that the area bounded by the parabola \(y=x^{2}\) and the line \(y=b(b>0)\) is two-thirds of the area of the rectangle that circumscribes it.

Find the coordinates of the centroids of the given figures. Each region is covered by a thin, flat plate. A highway marking pylon has the shape of a frustum of a cone. Find its centroid if the radii of its bases are \(5.00 \mathrm{cm}\) and \(20.0 \mathrm{cm}\) and the height between bases is \(60.0 \mathrm{cm}\).

Show that a \(1-\mathrm{m} \mathrm{A}\) constant current increases the voltage on a \(1-\mu \mathrm{F}\) capacitor by \(1 \mathrm{V}\) in \(1 \mathrm{ms}\)

The efficiency \(e\) (in \(\%\) ) of an automobile engine is given by \(e=0.768 s-0.00004 s^{3},\) where \(s\) is the speed (in \(\mathrm{km} / \mathrm{h}\) ) of the car. Find the average efficiency with respect to the speed for \(s=30.0 \mathrm{km} / \mathrm{h}\) to \(s=90.0 \mathrm{km} / \mathrm{h}\).

Solve the given problems. Show that the curve \(y=x^{n}(n>0)\) divides the unit square bounded by \(x=0, y=0, x=1,\) and \(y=1\) into regions with areas in the ratio of \(n / 1\)

The rate of change of the vertical deflection \(y\) with respect to the horizontal distance \(x\) from one end of a beam is a function of \(x\) For a particular beam, the function is \(k\left(x^{5}+1350 x^{3}-7000 x^{2}\right)\) where \(k\) is a constant. Find \(y\) as a function of \(x\) if \(y=0\) when \(x=0\)

Find the average value of the volume of a sphere with respect to the radius. Explain the meaning of the result.

Find the indicated volumes by integration. The oil in a spherical tank \(40.0 \mathrm{ft}\) in diameter is \(15.0 \mathrm{ft}\) deep. How much oil is in the tank?

Solve the given problems. Why can the integral \(\int_{a}^{2}\left(2+x-x^{2}\right) d x\) be used to find the area bounded by \(x=a, y=0,\) and \(y=2+x-x^{2}\) if \(a=-1,\) but not if \(a=-2 ?\)

Freshwater is flowing into a brine solution, with an equal volume of mixed solution flowing out. The amount of salt in the solution decreases, but more slowly as time increases. Under certain conditions, the time rate of change of mass of salt (in \(\mathrm{g} / \mathrm{min}\) ) is given by \(-1 / \sqrt{t+1}\). Find the mass \(m\) of salt as a function of time if 1000 g were originally present. Under these conditions, how long would it take for all the salt to be removed?

The length of arc \(s\) of a curve from \(x=a\) to \(x=b\) is $$s=\int_{a}^{b} \sqrt{1+\left(\frac{d y}{d x}\right)^{2}} d x$$ The cable of a bridge can be described by the equation \(y=0.04 x^{3 / 2}\) from \(x=0\) to \(x=100 \mathrm{ft}\). Find the length of the cable. See Fig. 26.67.

Solve the given problems. Find the area of the parallelogram with vertices at (0,0),(2,0) (2, 1), and (4, 1) by integration. Show any integrals you set up.

A holograph of a circle is formed. The rate of change of the radius \(r\) of the circle with respect to the wavelength \(\lambda\) of the light used is inversely proportional to the square root of \(\lambda\). If \(d r / d \lambda=3.55 \times 10^{4}\) and \(r=4.08 \mathrm{cm}\) for \(\lambda=574 \mathrm{nm},\) find \(r\) as a function of \(\lambda\)

Find the indicated volumes by integration. If the area bounded by \(y=0, y=2 x,\) and \(x=5\) is rotated about each axis, which volume is greater?

Solve the given problems. Set up the integrals (do not evaluate) for the upper of the two areas bounded by \(y=4-x^{2}, y=3 x,\) and \(y=4-2 x,\) using vertical elements of area.

Find the indicated volumes by integration. The ball used in Australian football is elliptical. Find its volume if it is \(275 \mathrm{mm}\) long and \(170 \mathrm{mm}\) wide.

The area of a surface of revolution from \(x=a\) to \(x=b\) is $$S=2 \pi \int_{a}^{b} y \sqrt{1+\left(\frac{d y}{d x}\right)^{2}} d x$$ Find the formula for the lateral surface area of a right circular cone of radius \(r\) and height \(h\).

Solve the given problems. Find the value of \(c\) such that the region bounded by \(y=x^{2}\) and \(y=4\) is divided by \(y=c\) into two regions of equal area.

The grinding surface of a grinding machine can be described as the surface generated by rotating the curve \(y=0.2 x^{3}\) from \(x=0\) to \(x=2.0 \mathrm{cm}\) about the \(x\) -axis. Find the grinding surface area. (See Exercise 37.)

Solve the given problems. Find the positive value of \(c\) such that the region bounded by \(y=x^{2}-c^{2}\) and \(y=c^{2}-x^{2}\) has an area of 576

Find the areas bounded by the indicated curves, using (a) vertical elements and (b) horizontal elements. $$y=8 x, x=0, y=4$$

Find the areas bounded by the indicated curves, using (a) vertical elements and (b) horizontal elements. $$y=x^{3}, x=0, y=3$$

Find the areas bounded by the indicated curves, using (a) vertical elements and (b) horizontal elements. $$y=4 x, y=x^{3}$$Find the areas bounded by the indicated curves, using (a) vertical elements and (b) horizontal elements. $$y=x^{4}, y=8 x$$

Find the areas bounded by the indicated curves, using (a) vertical elements and (b) horizontal elements. $$y=4 x, y=x^{3}$$