The accompanying figure shows a spherical cap of radius and height h cut from a sphere

A solid extends along the \(x\)-axis from \(x=1 \) to \(x=3\). For \(x\) between 1 and 3 , the cross-sectional area of \(S\) perpendicular to the \(x\)-axis is \(3 x^{2}\). An integral expression for the volume of \(S\) is ________, The value of this integral is ________. Equation Transcription: Text Transcription: x x=1 x=3 S 3x^2

A solid \(S\) is generated by revolving the region between the \(x\)-axis and the curve \(y=\sqrt{\sin x}(0 \leq x \leq \pi)\) about the \(x\) axis. (a) For \(x\) between 0 and \(\pi\), the cross-sectional area of perpendicular to the \(x\)-axis at \(x\) is \(A(x)=\) (b) An integral expression for the volume of \(S\) is (c) The value of the integral in part (b) is Equation Transcription: Text Transcription: S x y= square root sin x (0<=x<=pi) A(x)=

A solid is generated by revolving the region enclosed by the line \(y=2x+1\) and the curve \(y=x^{2}+1\) about the \(x\)-axis. (a) For x between and , the cross sectional area of \(S\) perpendicular to the \(x\)-axis at \(x\) is \(A(x) = \) (b) An integral expression for the volume of \(S\) is ._____ Equation Transcription: S A() = Text Transcription: y=2x+1 y=x^2+1 x S A(x) =

A solid \(S\) is generated by revolving the region enclosed by the line \(y=2x+1\) and the curve \(y=x^{2}+1\) about the \(x\)-axis. (a) For y between and , the cross sectional area of \(S\) perpendicular to the y-axis at y is \(A(y) =\). (b) An integral expression for the volume of \(S\) is . Equation Transcription: S Text Transcription: y=2x+1 y=x^2+1 x S A(y) =

18 Find the volume of the solid that results when the shaded region is revolved about the indicated axis. 5

18 Find the volume of the solid that results when the shaded region is revolved about the indicated axis. 6

18 Find the volume of the solid that results when the shaded region is revolved about the indicated axis. 7

18 Find the volume of the solid that results when the shaded region is revolved about the indicated axis. 8

Find the volume of the solid whose base is the region bounded between the curve \(y=x^{2}\) and the \(x\)-axis from \(x = 0\) to \(x = 2\) and whose cross sections taken perpendicular to the \(x\)-axis are squares. Equation Transcription: Text Transcription: y=x^2 x x = 0 x = 2

Find the volume of the solid whose base is the region bounded between the curve \(y = sec x\) and the \(x\)-axis from \(x=\pi / 4\) to \(x=\pi / 3\) and whose cross sections taken perpendicular to the \(x\)-axis are squares Equation Transcription: Text Transcription: y=sec x x x=pi4 x=pi/3

Find the volume of the solid that results when the region enclosed by the given curves is revolved about the x-axis. \(y=\sqrt{25-x^{2}}, y=3\) Equation Transcription: , Text Transcription: y = sqrt 25 - x^2 , y = 3

Find the volume of the solid that results when the region enclosed by the given curves is revolved about the x-axis. \(y=9-x^{2}, y=0\) Equation Transcription: Text Transcription: Y = 9 - x^2 , y = 0

Find the volume of the solid that results when the region enclosed by the given curves is revolved about the x-axis. \(x=\sqrt{y}, x=y / 4\) Equation Transcription: Text Transcription: X = sqrt y, x = y/4

Find the volume of the solid that results when the region enclosed by the given curves is revolved about the x-axis. \(y=\sin x, y=\cos x, x=0, x=\pi / 4\) [Hint: Use the identity \(\cos 2 x=\cos ^{2} x-\sin ^{2} x\).] Equation Transcription: Text Transcription: y = sinx , y = cosx, x = 0, x = pi/4 cos2x = cos^2 x - sin^2 x

Find the volume of the solid that results when the region enclosed by the given curves is revolved about the x-axis. \(y=e^{x}, y=0, x=0, x=\ln 3\) Equation Transcription: Text Transcription: Y = e^x , y = 0 , x = 0 , x = ln 3

Find the volume of the solid that results when the region enclosed by the given curves is revolved about the x-axis. \(y=e^{-2 x}, y=0, x=0, x=1\) Equation Transcription: Text Transcription: Y = e^-2x , y = 0, x = 0, x = 1

Find the volume of the solid that results when the region enclosed by the given curves is revolved about the x-axis. \(y=\frac{1}{\sqrt{4+x^{2}}}, x=-2, x=2, y=0\) Equation Transcription: Text Transcription: Y = 1/sqrt 4 + x^2 , x = -2, x = 2, y = 0

Find the volume of the solid that results when the region enclosed by the given curves is revolved about the x-axis. \(y=\frac{e^{3 x}}{\sqrt{1+e^{6 x}}}, x=0, x=1, y=0\) Equation Transcription: Text Transcription: Y = e^3x / sqrt 1 + e^6x , x = 0, x = 1, y = 0

Find the volume of the solid whose base is the region bounded between the curve \(y=x^{3}\) and the -axis from \(y=0\) to \(y=1\) and whose cross sections taken perpendicular to the -axis are squares. Equation Transcription: Text Transcription: y = x^3 y = 0 y = 1

Find the volume of the solid whose base is the region enclosed between the curve \(x=1-y^{2}\) and the -axis and whose cross sections taken perpendicular to the -axis are squares. Equation Transcription: Text Transcription: x = 1 - y^2

Find the volume of the solid that results when the region enclosed by the given curves is revolved about the -axis. \(x=c s c y, y=\pi / 4, y=3 \pi / 4, x=0\) Equation Transcription: Text Transcription: X = cscy, y = pi/4 , y = 3pi/4 , x = 0

Find the volume of the solid that results when the region enclosed by the given curves is revolved about the -axis. \(y=x^{2}, x=y^{2}\) Equation Transcription: Text Transcription: y = x^2 , x = y^2

Find the volume of the solid that results when the region enclosed by the given curves is revolved about the -axis. \(x=y^{2}, x=y+2\) Equation Transcription: Text Transcription: x = y^2 , x = y + 2

Find the volume of the solid that results when the region enclosed by the given curves is revolved about the -axis. \(x=1-y^{2}, x=2+y^{2}, y=-1, y=1\) Equation Transcription: Text Transcription: X = 1 - y^2 , x = 2 + y^2 , y = -1, y = 1

Find the volume of the solid that results when the region enclosed by the given curves is revolved about the -axis. \(y=\ln x, x=0, y=0, y=1\) Equation Transcription: Text Transcription: y = ln x, x = 0, y = 0, y = 1

Find the volume of the solid that results when the region enclosed by the given curves is revolved about the -axis. \(y=\sqrt{\frac{1-x^{2}}{x^{2}}}(x>0), x=0, y=0, y=2\) Equation Transcription: Text Transcription: Y = sqrt 1 - x^2 / x^2 (x>0) , x = 0, y = 0, y = 2

True-False Determine whether the statement is true or false. Explain your answer. [In these exercises, assume that a solid of volume is bounded by two parallel planes perpendicular to the -axis at \(x=a\) and \(x=b\) and that for each in \([a, b], A(x)\) denotes the cross-sectional area of perpendicular to the -axis.] If each cross section of perpendicular to the -axis is a square, then is a rectangular parallelepiped (i.e., is box shaped). Equation Transcription: Text Transcription: X = a X = b [a,b], A (x)

True-False Determine whether the statement is true or false. Explain your answer. [In these exercises, assume that a solid of volume is bounded by two parallel planes perpendicular to the -axis at \(x=a\) and \(x=b\) and that for each in [\(a, b], A(x)\) denotes the cross-sectional area of perpendicular to the -axis.] If each cross section of is a disk or a washer, then is a solid of revolution. Equation Transcription: Text Transcription: x = a x = b [a,b], A (x)

True-False Determine whether the statement is true or false. Explain your answer. [In these exercises, assume that a solid of volume is bounded by two parallel planes perpendicular to the -axis at \(x=a\) and \(x=b\) and that for each in \([a, b], A(x)\) denotes the cross-sectional area of perpendicular to the -axis.] If is in centimeters , then \(A(x)\) must be a quadratic function of , since units of \(A(x)\) will be square centimeters \(\left(\mathrm{cm}^{2}\right)\) Equation Transcription: Text Transcription: X = a X = b [a,b], A(x) A (x) cm^2

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

Find the volume of the solid that results when the region above the -axis and below the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>0, b>0)\) is revolved about the -axis. Equation Transcription: += Text Transcription: x^2 /a^2 + y^2 / b^2 = 1 (a > 0 , b > 0)

Let \(V\) be the volume of the solid that results when the region enclosed by \(y=1 / x, y=0, x=2\), and \(x=b(0<b<2)\) is revolved about the -axis. Find the value of \(b\) for which \(V=3\) Equation Transcription: Text Transcription: V y=1/x , y=0, x=2 x= b (0 < b < 2) b V = 3

Find the volume of the solid generated when the region enclosed by \(y=\sqrt{x+1}, y=\sqrt{2 x}\) , and \(y=0\) is revolved about the -axis. [Hint: Split the solid into two parts.] Equation Transcription: Text Transcription: y = sqrt x + 1, y = sqrt 2x y = 0

Find the volume of the solid generated when the region enclosed by \(y=\sqrt{x}, y=6-x\) , and \(y=0\) is revolved about the -axis. [Hint: Split the solid into two parts.] Equation Transcription: Text Transcription: y = sqrt x , y = 6 - x , y = 0

Suppose that \(f\) is a continuous function on \([a,b]\), and let \(R\) be the region between the curve \(y=f(x)\) and the line \(y=k\) from \(x=a\) to \(x=b\). Using the method of disks, derive with explanation a formula for the volume of a solid generated by revolving \(R\) about the line \(y=k\). State and explain additional assumptions, if any, that you need about \(f\) for your formula. Equation Transcription: Text Transcription: f [a,b] R y = f(x) y = k x = a x = b

Suppose that \(v\) and \(w\) are continuous functions on \([c,d]\), and let \(R\) be the region between the curves \(x=v(y)\) and \(x=w(y)\) from \(y=c\) to \(y=d\). Using the method of washers, derive with explanation a formula for the volume of a solid generated by revolving \(R\) about the line \(x=k\). State and explain additional assumptions, if any, that you need about v and w for your formula. Equation Transcription: Text Transcription: v w [c,d] R x = v(y) x = w(y)

Consider the solid generated by revolving the shaded region in Exercise 1 about the line \(y=2\). (a) Make a conjecture as to which is larger: the volume of this solid or the volume of the solid in Exercise 1. Explain the basis of your conjecture. (b) Check your conjecture by calculating this volume and comparing it to the volume obtained in Exercise 1 . Equation Transcription: Text Transcription: y = 2

Consider the solid generated by revolving the shaded region in Exercise 4 about the line \(x=2.5\). (a) Make a conjecture as to which is larger: the volume of this solid or the volume of the solid in Exercise 4. Explain the basis of your conjecture. (b) Check your conjecture by calculating this volume and comparing it to the volume obtained in Exercise 4 . Equation Transcription: Text Transcription: x = 2.5

Find the volume of the solid that results when the region enclosed by \(y=\sqrt{x}, y=0\), and \(x=9\) is revolved about the line \(x=9\) Equation Transcription: Text Transcription: Y = sqrt x , y = 0 , x = 9 x = 9

Find the volume of the solid that results when the region in Exercise 39 is revolved about the line \(y=3\) . Equation Transcription: Text Transcription: y = 3

Find the volume of the solid that results when the region enclosed by \(x=y^{2}\) and \(x=y\) is revolved about the line \(y=-1\). Equation Transcription: x = y2 x = y y = -1 Text Transcription: x = y^2 x = y y = -1

Find the volume of the solid that results when the region in Exercise \(41\) is revolved about the line \(x = -1\). Equation Transcription: 41 x = -1 Text Transcription: 41 x = -1

Find the volume of the solid that results when the region enclosed by \(y=x^{2}\) and \(y=x^{3}\) is revolved about the line \(x=1\). Equation Transcription: y = x2 y = x3 x = 1 Text Transcription: y = x^2 y = x^3 x = 1

Find the volume of the solid that results when the region in Exercise 43 is revolved about the line \(y=-1\) Equation Transcription: Text Transcription: y = -1

A nose cone for a space reentry vehicle is designed so that a cross section, taken \(xft\) from the tip and perpendicular to the axis of symmetry, is a circle of radius \(\frac{1}{4} x^{2} f t\). Find the volume of the nose cone given that its length is \(20ft\). Equation Transcription: Text Transcription: xft 1/4 x^2 ft 20 ft

A certain solid is \(1ft\) high, and a horizontal cross section taken \(xft\) above the bottom of the solid is an annulus of inner radius \(x^{2} f t\) and outer radius \(\sqrt{x} f t\) . Find the volume of the solid. Equation Transcription: Text Transcription: 1 ft xft x^2 ft Sqrt xft

Find the volume of the solid whose base is the region bounded between the curves \(y=x\) and \(y=x^{2}\), and whose cross sections perpendicular to the -axis are squares. Equation Transcription: Text Transcription: y = x y = x^2

The base of a certain solid is the region enclosed by \(y=\sqrt{x}\), \(y=0\), and \(x=4\). Every cross section perpendicular to the -axis is a semicircle with its diameter across the base. Find the volume of the solid. Equation Transcription: Text Transcription: y = sqrt x , y = 0 x = 4

In parts (a)–(c) find the volume of the solid whose base is enclosed by the circle \(x^{2}+y^{2}=1\) and whose cross sections taken perpendicular to the x-axis are (a) semicircles (b) squares (c) equilateral triangles. Equation Transcription: Text Transcription: x^2 + y^2 = 1

As shown in the accompanying figure, a cathedral dome is designed with three semicircular supports of radius \(r\) so that each horizontal cross section is a regular hexagon. Show that the volume of the dome is \(r^{3} \sqrt{3}\). Equation Transcription: Text Transcription: r r^3 sqrt 3

Use a CAS to estimate the volume of the solid that results when the region enclosed by the curves is revolved about the stated axis. \(y=\sin ^{8} x, \ y=2 x / \pi, \ x=0, \ x=\pi / 2 ; \ x \text {-axis }\) Equation Transcription: y = sin8?x, y = 2x/, x = 0, x = /2; x-axis Text Transcription: y = sin^8?x, y = 2x/pi, x = 0, x = pi/2; x-axis

Use a CAS to estimate the volume of the solid that results when the region enclosed by the curves is revolved about the stated axis. \(y=\pi^{2} \sin x \cos ^{3} x, \ y=4 x^{2}, \ x=0, \ x=\pi / 4 ; \ x \text {-axis }\) Equation Transcription: y = 2 sin? x cos3?x, y = 4x2, x = 0, x = /4; x-axis Text Transcription: y = pi^2 sin? x cos^3 ?x, y = 4x^2, x = 0, x = pi/4; x-axis

Use a CAS to estimate the volume of the solid that results when the region enclosed by the curves is revolved about the stated axis. \(y=e^{x}, \ x=1, \ y=1 ; \ y-\text { axis }\) Equation Transcription: y=ex, x = 1, y = 1; y-axis Text Transcription: y=e^x, x = 1, y = 1; y-axis

Use a CAS to estimate the volume of the solid that results when the region enclosed by the curves is revolved about the stated axis. \(y=x \sqrt{\tan ^{-1} x}, \ y=x ; \ x \text {-axis }\) Equation Transcription: y = x , y = x; x-axis Text Transcription: y = x sqrt tan^-1 x, y = x; x-axis

The accompanying figure shows a spherical cap of radius \(\rho\) and height \(h\) cut from a sphere of radius \(r\). Show that the volume \(V\) of the spherical cap can be expressed as (a) \(V=\frac{1}{3} \pi h^{2}(3 r-h)\) (b) \(V=\frac{1}{6} \pi h\left(3 \rho^{2}+h^{2}\right)\) Equation Transcription: h r V V = h2(3r - h) V = h2(3) Text Transcription: rho h r V V = 1/6 pi h^2(3r - h) V = 1/6 pi h^2(3rho^2 + h^2)

If fluid enters a hemispherical bowl with a radius of \(10 ft\) at a rate of \(\frac{1}{2} \ f t^{3} / \min\), how fast will the fluid be rising when the depth is \(5 ft\)? [Hint: See Exercise \(5.5\)] Equation Transcription: 10 ft ft3/min 5 ft 5.5 Text Transcription: 10 ft 1/2 ft^3/min 5 ft 5.5

The accompanying figure (on the next page) shows the dimensions of a small lightbulb at \(10\) equally spaced points. (a) Use formulas from geometry to make a rough estimate of the volume enclosed by the glass portion of the bulb. (b) Use the average of left and right endpoint approximations to approximate the volume. Equation Transcription: 10 Text Transcription: 10

Use the result in Exercise \(55\) to find the volume of the solid that remains when a hole of radius \(r/2\) is drilled through the center of a sphere of radius \(r\), and then check your answer by integrating. Equation Transcription: 55 r/2 r Text Transcription: 55 r/2 r

As shown in the accompanying figure, a cocktail glass with a bowl shaped like a hemisphere of diameter \(8 \ cm\) contains a cherry with a diameter of \(2 \ cm\). If the glass is filled to a depth of \(h \ cm\), what is the volume of liquid it contains? [Hint: First consider the case where the cherry is partially submerged, then the case where it is totally submerged.] Equation Transcription: 8 cm 2 cm h cm Text Transcription: 8 cm 2 cm h cm

Find the volume of the torus that results when the region enclosed by the circle of radius \(r\) with center at \((h, 0), \ h>r\), is revolved about the y-axis. [Hint: Use an appropriate formula from plane geometry to help evaluate the definite integral.] Equation Transcription: r (h, 0), h > r Text Transcription: r (h, 0), h > r

A wedge is cut from a right circular cylinder of radius \(r\) by two planes, one perpendicular to the axis of the cylinder and the other making an angle \(\theta\) with the first. Find the volume of the wedge by slicing perpendicular to the y-axis as shown in the accompanying figure. Equation Transcription: r Text Transcription: r theta

Find the volume of the wedge described in Exercise \(61\) by slicing perpendicular to the x-axis. Equation Transcription: 61 Text Transcription: 61

Two right circular cylinders of radius \(r\) have axes that intersect at right angles. Find the volume of the solid common to the two cylinders. [Hint: One-eighth of the solid is sketched in the accompanying figure.] Equation Transcription: r Text Transcription: r

In \(1635\) Bonaventura Cavalieri, a student of Galileo, stated the following result, called Cavalieri’s principle: If two solids have the same height, and if the areas of their cross sections taken parallel to and at equal distances from their bases are always equal, then the solids have the same volume. Use this result to find the volume of the oblique cylinder in the accompanying figure. (See Exercise \(52\) of Section \(6.1\) for a planar version of Cavalieri’s principle.) Equation Transcription: 1635 52 6.1 Text Transcription: 1635 52 6.1

Use the results of this section to derive Cavalieri’s principle (Exercise \(64\)). Equation Transcription: 64 Text Transcription: 64

Write a short paragraph that explains how Formulas \((4)–(8)\) may all be viewed as consequences of Formula \((3)\). Equation Transcription: (4)–(8) (3) Text Transcription: (4)–(8) (3)