Example: Arc Length and Surface Area

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Example: Arc Length and Surface Area

Find the arc length or surface area of a function over a span on which its derivative is continuous.

Arc Length

1. Define a function of one variable.

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2. Define the endpoints of an interval.

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3. Plot the function f over the interval [a, b] by setting the lowest tick mark value to a and the highest tick mark value to b.

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<region id="ID0EQ5XFB" top="312" left="24" height="312" width="312" xmlns="http://schemas.mathsoft.com/worksheet50">
  <plot origin-positioning="true">
    <xyPlot>
      <title class="- topic/title " toc="yes" wwtype:type="Paragraph" xmlns:wwtype="urn:WebWorks-Type-Schema" />
      <legend />
      <traces>
        <trace num-of-points="500">
          <traceStyle color="#FF00008B" symbol="none" line-weight="1" line-style="Solid">lines</traceStyle>
        </trace>
      </traces>
      <axes>
        <xAxis rank="1" legend-position="PlotBoundaryBottom">
          <axisLine position="origin" positionticmark="5" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat />
          <plotEquations>
            <plotEquation>
              <math>
                <id xmlns="http://schemas.mathsoft.com/math50">x</id>
              </math>
              <math>
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
          </plotEquations>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue>
              <id xmlns="http://schemas.mathsoft.com/math50">a</id>
            </startValue>
            <endValue>
              <id xmlns="http://schemas.mathsoft.com/math50">b</id>
            </endValue>
          </xyDomain>
        </xAxis>
        <yAxis rank="1" legend-position="PlotBoundaryLeft">
          <axisLine position="origin" positionticmark="0" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat />
          <plotEquations>
            <plotEquation>
              <math>
                <apply xmlns="http://schemas.mathsoft.com/math50">
                  <id>f</id>
                  <id>x</id>
                </apply>
              </math>
              <math>
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
          </plotEquations>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <endValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </endValue>
          </xyDomain>
        </yAxis>
      </axes>
    </xyPlot>
  </plot>
</region>

4. Calculate the arc length over the interval [a, b].

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<region id="ID0EFAYFB" top="744" left="48" height="92.164" width="180.161315865318" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math>
    <eval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <integral />
        <lambda>
          <boundVars>
            <id>x</id>
          </boundVars>
          <apply>
            <nthRoot />
            <placeholder />
            <apply>
              <plus />
              <real>1</real>
              <apply>
                <pow />
                <parens>
                  <apply>
                    <derivative />
                    <lambda>
                      <boundVars>
                        <id>x</id>
                      </boundVars>
                      <apply>
                        <id>f</id>
                        <id>x</id>
                      </apply>
                    </lambda>
                    <degree>
                      <placeholder />
                    </degree>
                  </apply>
                </parens>
                <real>2</real>
              </apply>
            </apply>
          </apply>
        </lambda>
        <lowerBound>
          <id>a</id>
        </lowerBound>
        <upperBound>
          <id>b</id>
        </upperBound>
      </apply>
      <result>
        <real>12.056</real>
      </result>
    </eval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" />
    </resultFormat>
  </math>
</region>

Surface Area

Calculate the surface area above a disc as described by the following condition:

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1. Define a function of two variables.

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2. Define a radius for the disc-shaped region of integration.

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3. Calculate the surface area above the disc.

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<region id="ID0EYFYFB" top="1248" left="48" height="110.551800578833" width="325.3466491355" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math>
    <eval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <integral />
        <lambda>
          <boundVars>
            <id>x</id>
          </boundVars>
          <apply>
            <integral />
            <lambda>
              <boundVars>
                <id>y</id>
              </boundVars>
              <apply>
                <nthRoot />
                <placeholder />
                <apply>
                  <plus />
                  <apply>
                    <plus />
                    <real>1</real>
                    <apply>
                      <pow />
                      <parens>
                        <apply>
                          <derivative />
                          <lambda>
                            <boundVars>
                              <id>x</id>
                            </boundVars>
                            <apply>
                              <id>f</id>
                              <sequence>
                                <id>x</id>
                                <id>y</id>
                              </sequence>
                            </apply>
                          </lambda>
                          <degree>
                            <placeholder />
                          </degree>
                        </apply>
                      </parens>
                      <real>2</real>
                    </apply>
                  </apply>
                  <apply>
                    <pow />
                    <parens>
                      <apply>
                        <derivative />
                        <lambda>
                          <boundVars>
                            <id>y</id>
                          </boundVars>
                          <apply>
                            <id>f</id>
                            <sequence>
                              <id>x</id>
                              <id>y</id>
                            </sequence>
                          </apply>
                        </lambda>
                        <degree>
                          <placeholder />
                        </degree>
                      </apply>
                    </parens>
                    <real>2</real>
                  </apply>
                </apply>
              </apply>
            </lambda>
            <lowerBound>
              <apply>
                <neg />
                <apply>
                  <nthRoot />
                  <placeholder />
                  <apply>
                    <minus />
                    <apply>
                      <pow />
                      <id>r</id>
                      <real>2</real>
                    </apply>
                    <apply>
                      <pow />
                      <id>x</id>
                      <real>2</real>
                    </apply>
                  </apply>
                </apply>
              </apply>
            </lowerBound>
            <upperBound>
              <apply>
                <nthRoot />
                <placeholder />
                <apply>
                  <minus />
                  <apply>
                    <pow />
                    <id>r</id>
                    <real>2</real>
                  </apply>
                  <apply>
                    <pow />
                    <id>x</id>
                    <real>2</real>
                  </apply>
                </apply>
              </apply>
            </upperBound>
          </apply>
        </lambda>
        <lowerBound>
          <apply>
            <neg />
            <id>r</id>
          </apply>
        </lowerBound>
        <upperBound>
          <id>r</id>
        </upperBound>
      </apply>
      <result>
        <real>13.614</real>
      </result>
    </eval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" />
    </resultFormat>
  </math>
</region>

4. Convert the function into polar coordinates and calculate the surface area above the disc.

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<region id="ID0EZHYFB" top="1464" left="48" height="99.5330005123615" width="313.209982468834" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math>
    <eval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <integral />
        <lambda>
          <boundVars>
            <id>θ</id>
          </boundVars>
          <apply>
            <integral />
            <lambda>
              <boundVars>
                <id>r</id>
              </boundVars>
              <apply>
                <scale />
                <apply>
                  <nthRoot />
                  <placeholder />
                  <apply>
                    <plus />
                    <apply>
                      <plus />
                      <real>1</real>
                      <apply>
                        <pow />
                        <parens>
                          <apply>
                            <derivative />
                            <lambda>
                              <boundVars>
                                <id>r</id>
                              </boundVars>
                              <apply>
                                <id>f</id>
                                <sequence>
                                  <id>r</id>
                                  <id>θ</id>
                                </sequence>
                              </apply>
                            </lambda>
                            <degree>
                              <placeholder />
                            </degree>
                          </apply>
                        </parens>
                        <real>2</real>
                      </apply>
                    </apply>
                    <apply>
                      <pow />
                      <parens>
                        <apply>
                          <derivative />
                          <lambda>
                            <boundVars>
                              <id>θ</id>
                            </boundVars>
                            <apply>
                              <id>f</id>
                              <sequence>
                                <id>r</id>
                                <id>θ</id>
                              </sequence>
                            </apply>
                          </lambda>
                          <degree>
                            <placeholder />
                          </degree>
                        </apply>
                      </parens>
                      <real>2</real>
                    </apply>
                  </apply>
                </apply>
                <id>r</id>
              </apply>
            </lambda>
            <lowerBound>
              <real>0</real>
            </lowerBound>
            <upperBound>
              <apply>
                <nthRoot />
                <placeholder />
                <real>2</real>
              </apply>
            </upperBound>
          </apply>
        </lambda>
        <lowerBound>
          <real>0</real>
        </lowerBound>
        <upperBound>
          <apply>
            <scale />
            <real>2</real>
            <id>π</id>
          </apply>
        </upperBound>
      </apply>
      <result>
        <real>13.614</real>
      </result>
    </eval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" />
    </resultFormat>
  </math>
</region>