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Mathematics LibreTexts

6.4E: Exercises for Section 6.4

  • Page ID
    34065
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    For exercises 1 - 3, find the length of the functions over the given interval.

    1) \( y=5x\) from \( x=0\) to \( x=2\)

    Answer:
    \(s = 2\sqrt{26}\) units

    2) \( y=−\frac{1}{2}x+25\) from \( x=1\) to \( x=4\)

    3) \( x=4y\) from \( y=−1\) to \( y=1\)

    Answer:
    \( s = 2\sqrt{17}\) units

     

    4) Pick an arbitrary linear function \( x=g(y)\) over any interval of your choice \( (y_1,y_2).\) Determine the length of the function and then prove the length is correct by using geometry.

    5) Find the surface area of the volume generated when the curve \( y=\sqrt{x}\) revolves around the \(x\)-axis from \( (1,1)\) to \( (4,2)\), as seen here.

    Answer:
    \(A = \frac{π}{6}(17\sqrt{17}−5\sqrt{5})\) units2

    6) Find the surface area of the volume generated when the curve \( y=x^2\) revolves around the \(y\)-axis from \( (1,1)\) to \( (3,9)\).

     

    For exercises 7 - 16, find the lengths of the functions of \(x\) over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it.

    7) \( y=x^{3/2}\) from \( (0,0)\) to \( (1,1)\)

    Answer:
    \(s= \frac{13\sqrt{13}−8}{27}\) units

    8) \( y=x^{2/3}\) from \( (1,1)\) to \( (8,4)\)

    9) \( y=\frac{1}{3}(x^2+2)^{3/2}\) from \( x=0\) to \( x=1\)

    Answer:
    \(s= \frac{4}{3}\) units

    10) \( y=\frac{1}{3}(x^2−2)^{3/2}\) from \( x=2\) to \( x=4\)

    11) [T] \( y=e^x\) on \( x=0\) to \( x=1\)

    Answer:
    \(s \approx 2.0035\) units

    12) \( y=\dfrac{x^3}{3}+\dfrac{1}{4x}\) from \( x=1\) to \( x=3\)

    13) \( y=\dfrac{x^4}{4}+\dfrac{1}{8x^2}\) from \( x=1\) to \( x=2\)

    Answer:
    \(s= \frac{123}{32}\) units

    14) \( y=\dfrac{2x^{3/2}}{3}−\dfrac{x^{1/2}}{2}\) from \( x=1\) to \( x=4\)

    15) \( y=\frac{1}{27}(9x^2+6)^{3/2}\) from \( x=0\) to \( x=2\)

    Answer:
    \(s=10\) units

    16) [T] \( y=\sin x\) on \( x=0\) to \( x=π\)

     

    For exercises 17 - 26, find the lengths of the functions of \(y\) over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it.

    17) \( y=\dfrac{5−3x}{4}\) from \( y=0\) to \( y=4\)

    Answer:
    \(s= \frac{20}{3}\) units

    18) \( x=\frac{1}{2}(e^y+e^{−y})\) from \( y=−1\) to \( y=1\)

    19) \( x=5y^{3/2}\) from \( y=0\) to \( y=1\)

    Answer:
    \(s= \frac{1}{675}(229\sqrt{229}−8)\) units

    20) [T] \( x=y^2\) from \( y=0\) to \( y=1\)

    21) \( x=\sqrt{y}\) from \( y=0\) to \( y=1\)

    Answer:
    \(s= \frac{1}{8}(4\sqrt{5}+\ln(9+4\sqrt{5}))\) units

    22) \( x=\frac{2}{3}(y^2+1)^{3/2}\) from \( y=1\) to \( y=3\)

    23) [T] \( x=\tan y\) from \( y=0\) to \( y=\frac{3}{4}\)

    Answer:
    \(s \approx 1.201\) units

    24) [T] \( x=\cos^2y\) from \( y=−\frac{π}{2}\) to \( y=\frac{π}{2}\)

    25) [T] \( x=4^y\) from \( y=0\) to \( y=2\)

    Answer:
    \(s \approx 15.2341\) units

    26) [T] \( x=\ln(y)\) on \( y=\dfrac{1}{e}\) to \( y=e\)

     

    For exercises 27 - 34, find the surface area of the volume generated when the following curves revolve around the \(x\)-axis. If you cannot evaluate the integral exactly, use your calculator to approximate it.

    27) \( y=\sqrt{x}\) from \( x=2\) to \( x=6\)

    Answer:
    \(A= \frac{49π}{3}\) units2

    28) \( y=x^3\) from \( x=0\) to \( x=1\)

    29) \( y=7x\) from \( x=−1\) to \( x=1\)

    Answer:
    \(A = 70π\sqrt{2}\) units2

    30) [T] \( y=\frac{1}{x^2}\) from \( x=1\) to \( x=3\)

    31) \( y=\sqrt{4−x^2}\) from \( x=0\) to \( x=2\)

    Answer:
    \(A = 8π\) units2

    32) \( y=\sqrt{4−x^2}\) from \( x=−1\) to \( x=1\)

    33) \( y=5x\) from \( x=1\) to \( x=5\)

    Answer:
    \(A = 120π\sqrt{26}\) units2

    34) [T] \( y=\tan x\) from \( x=−\frac{π}{4}\) to \( x=\frac{π}{4}\)

     

    For exercises 35 - 42, find the surface area of the volume generated when the following curves revolve around the \(y\)-axis. If you cannot evaluate the integral exactly, use your calculator to approximate it.

    35) \( y=x^2\) from \( x=0\) to \( x=2\)

    Answer:
    \(A= \frac{π}{6}(17\sqrt{17}−1)\) units2

    36) \( y=\frac{1}{2}x^2+\frac{1}{2}\) from \( x=0\) to \( x=1\)

    37) \( y=x+1\) from \( x=0\) to \( x=3\)

    Answer:
    \(A = 9\sqrt{2}π\) units2

    38) [T] \( y=\dfrac{1}{x}\) from \( x=\dfrac{1}{2}\) to \( x=1\)

    39) \( y=\sqrt[3]{x}\) from \( x=1\) to \( x=27\)

    Answer:
    \(A = \frac{10\sqrt{10}π}{27}(73\sqrt{73}−1)\) units2

    40) [T] \( y=3x^4\) from \( x=0\) to \( x=1\)

    41) [T] \( y=\dfrac{1}{\sqrt{x}}\) from \( x=1\) to \( x=3\)

    Answer:
    \(A \approx 25.645\) units2

    42) [T] \( y=\cos x\) from \( x=0\) to \( x=\frac{π}{2}\)

     

    43) The base of a lamp is constructed by revolving a quarter circle \( y=\sqrt{2x−x^2}\) around the \(y\)-axis from \( x=1\) to \( x=2\), as seen here. Create an integral for the surface area of this curve and compute it.

    Answer:
    \(A = 2π\) units2

    44) A light bulb is a sphere with radius \(1/2\) in. with the bottom sliced off to fit exactly onto a cylinder of radius \(1/4\) in. and length \(1/3\) in., as seen here. The sphere is cut off at the bottom to fit exactly onto the cylinder, so the radius of the cut is \(1/4\) in. Find the surface area (not including the top or bottom of the cylinder).

    45) [T] A lampshade is constructed by rotating \( y=1/x\) around the \(x\)-axis from \( y=1\) to \( y=2\), as seen here. Determine how much material you would need to construct this lampshade—that is, the surface area—accurate to four decimal places.

    Answer:
    \(10.5017\) units2

    46) [T] An anchor drags behind a boat according to the function \( y=24e^{−x/2}−24\), where \( y\) represents the depth beneath the boat and \( x\) is the horizontal distance of the anchor from the back of the boat. If the anchor is \( 23\) ft below the boat, how much rope do you have to pull to reach the anchor? Round your answer to three decimal places.

    47) [T] You are building a bridge that will span \( 10\) ft. You intend to add decorative rope in the shape of \( y=5|\sin((xπ)/5)|\), where \( x\) is the distance in feet from one end of the bridge. Find out how much rope you need to buy, rounded to the nearest foot.

    Answer:
    \( 23\) ft

     

    For exercise 48, find the exact arc length for the following problems over the given interval.

    48) \( y=\ln(\sin x)\) from \( x=π/4\) to \( x=(3π)/4\). (Hint: Recall trigonometric identities.)

     

    49) Draw graphs of \(y=x^2, y=x^6\), and \(y=x^{10}\). For \( y=x^n\), as \( n\) increases, formulate a prediction on the arc length from \( (0,0)\) to \( (1,1)\). Now, compute the lengths of these three functions and determine whether your prediction is correct.

    Answer:
    \(2\)

    50) Compare the lengths of the parabola \(x=y^2\) and the line \(x=by\) from \((0,0)\) to \((b^2,b)\) as \(b\) increases. What do you notice?

    51) Solve for the length of \(x=y^2\) from \((0,0)\) to \((1,1)\). Show that \( x=\dfrac{y^2}{2}\) from \((0,0)\) to \((2,2)\) is twice as long. Graph both functions and explain why this is so.

    Answer:
    Answers may vary

    52) [T] Which is longer between \((1,1)\) and \(\left(2,\frac{1}{2}\right)\): the hyperbola \(y=\dfrac{1}{x}\) or the graph of \(x+2y=3\)?

    53) Explain why the surface area is infinite when \(y=1/x\) is rotated around the \(x\)-axis for \( 1≤x<∞,\) but the volume is finite.

    Answer:
    For more information, look up Gabriel’s Horn.