範例：弧線長度與曲面面積

繪圖 > XY 軸座標圖 > 繪製函數 > 範例：弧線長度與曲面面積

範例：弧線長度與曲面面積

針對合併區中其導數為連續的函數，求解其弧長或曲面面積。

弧長

1. 定義含有一個變數的函數。

按一下以複製此運算式

<region id="ID0EV1VMB" top="168" left="48" height="21.2" width="136.66" xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math xml:lang="zh_TW">
    <define xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50">
      <function xml:lang="zh_TW">
        <id xml:lang="zh_TW" label="None">f</id>
        <boundVars xml:lang="zh_TW">
          <id xml:lang="zh_TW" label="None">x</id>
        </boundVars>
      </function>
      <apply xml:lang="zh_TW">
        <id xml:lang="zh_TW">sin</id>
        <apply xml:lang="zh_TW">
          <plus xml:lang="zh_TW" />
          <id xml:lang="zh_TW">x</id>
          <apply xml:lang="zh_TW">
            <mult xml:lang="zh_TW" />
            <id xml:lang="zh_TW">x</id>
            <apply xml:lang="zh_TW">
              <id xml:lang="zh_TW">sin</id>
              <id xml:lang="zh_TW">x</id>
            </apply>
          </apply>
        </apply>
      </apply>
    </define>
    <resultFormat xml:lang="zh_TW">
      <matrix size="12,12" offset="0,0" xml:lang="zh_TW" />
    </resultFormat>
  </math>
</region>

2. 定義一個區間的端點。

按一下以複製此運算式

按一下以複製此運算式

3. 將刻度線值下限設定為 a，並將刻度線值上限設定為 b，在區間 [a, b] 中繪製函數 f。

按一下以複製此運算式

<region id="ID0EF6VMB" top="312" left="24" height="312" width="312" xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/worksheet50">
  <plot origin-positioning="true" xml:lang="zh_TW">
    <xyPlot xml:lang="zh_TW">
      <title class="- topic/title " xml:lang="zh_TW" toc="yes" wwtype:type="Paragraph" xmlns:wwtype="urn:WebWorks-Type-Schema" />
      <legend xml:lang="zh_TW" />
      <traces xml:lang="zh_TW">
        <trace xml:lang="zh_TW" num-of-points="500">
          <traceStyle color="#FF00008B" symbol="none" line-weight="1" line-style="Solid" xml:lang="zh_TW">線條</traceStyle>
        </trace>
      </traces>
      <axes xml:lang="zh_TW">
        <xAxis rank="1" legend-position="PlotBoundaryBottom" xml:lang="zh_TW">
          <axisLine position="origin" positionticmark="5" xml:lang="zh_TW" />
          <axisGrid xml:lang="zh_TW">
            <gridFrequency xml:lang="zh_TW">11</gridFrequency>
            <gridLabels display="true" xml:lang="zh_TW" />
            <gridLines xml:lang="zh_TW" />
            <tickMarks display="true" xml:lang="zh_TW" />
          </axisGrid>
          <axisLabel xml:lang="zh_TW" />
          <markers xml:lang="zh_TW" />
          <numberFormat xml:lang="zh_TW" />
          <plotEquations xml:lang="zh_TW">
            <plotEquation xml:lang="zh_TW">
              <math xml:lang="zh_TW">
                <id xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50">x</id>
              </math>
              <math xml:lang="zh_TW">
                <placeholder xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
          </plotEquations>
          <xyDomain scale-type="linear" auto-scale="true" xml:lang="zh_TW">
            <startValue xml:lang="zh_TW">
              <id xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50">a</id>
            </startValue>
            <endValue xml:lang="zh_TW">
              <id xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50">b</id>
            </endValue>
          </xyDomain>
        </xAxis>
        <yAxis rank="1" legend-position="PlotBoundaryLeft" xml:lang="zh_TW">
          <axisLine position="origin" positionticmark="0" xml:lang="zh_TW" />
          <axisGrid xml:lang="zh_TW">
            <gridFrequency xml:lang="zh_TW">11</gridFrequency>
            <gridLabels display="true" xml:lang="zh_TW" />
            <gridLines xml:lang="zh_TW" />
            <tickMarks display="true" xml:lang="zh_TW" />
          </axisGrid>
          <axisLabel xml:lang="zh_TW" />
          <markers xml:lang="zh_TW" />
          <numberFormat xml:lang="zh_TW" />
          <plotEquations xml:lang="zh_TW">
            <plotEquation xml:lang="zh_TW">
              <math xml:lang="zh_TW">
                <apply xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50">
                  <id xml:lang="zh_TW">f</id>
                  <id xml:lang="zh_TW">x</id>
                </apply>
              </math>
              <math xml:lang="zh_TW">
                <placeholder xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
          </plotEquations>
          <xyDomain scale-type="linear" auto-scale="true" xml:lang="zh_TW">
            <startValue xml:lang="zh_TW">
              <placeholder xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <endValue xml:lang="zh_TW">
              <placeholder xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50" />
            </endValue>
          </xyDomain>
        </yAxis>
      </axes>
    </xyPlot>
  </plot>
</region>

4. 計算區間 [a, b] 中的弧長。

按一下以複製此運算式

<region id="ID0E6AWMB" top="744" left="48" height="92.164" width="180.161315865318" xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math xml:lang="zh_TW">
    <eval xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50">
      <apply xml:lang="zh_TW">
        <integral xml:lang="zh_TW" />
        <lambda xml:lang="zh_TW">
          <boundVars xml:lang="zh_TW">
            <id xml:lang="zh_TW">x</id>
          </boundVars>
          <apply xml:lang="zh_TW">
            <nthRoot xml:lang="zh_TW" />
            <placeholder xml:lang="zh_TW" />
            <apply xml:lang="zh_TW">
              <plus xml:lang="zh_TW" />
              <real xml:lang="zh_TW">1</real>
              <apply xml:lang="zh_TW">
                <pow xml:lang="zh_TW" />
                <parens xml:lang="zh_TW">
                  <apply xml:lang="zh_TW">
                    <derivative xml:lang="zh_TW" />
                    <lambda xml:lang="zh_TW">
                      <boundVars xml:lang="zh_TW">
                        <id xml:lang="zh_TW">x</id>
                      </boundVars>
                      <apply xml:lang="zh_TW">
                        <id xml:lang="zh_TW">f</id>
                        <id xml:lang="zh_TW">x</id>
                      </apply>
                    </lambda>
                    <degree xml:lang="zh_TW">
                      <placeholder xml:lang="zh_TW" />
                    </degree>
                  </apply>
                </parens>
                <real xml:lang="zh_TW">2</real>
              </apply>
            </apply>
          </apply>
        </lambda>
        <lowerBound xml:lang="zh_TW">
          <id xml:lang="zh_TW">a</id>
        </lowerBound>
        <upperBound xml:lang="zh_TW">
          <id xml:lang="zh_TW">b</id>
        </upperBound>
      </apply>
      <result xml:lang="zh_TW">
        <real xml:lang="zh_TW">12.056</real>
      </result>
    </eval>
    <resultFormat xml:lang="zh_TW">
      <matrix size="12,12" offset="0,0" xml:lang="zh_TW" />
    </resultFormat>
  </math>
</region>

曲面面積

如下列條件所述，計算圓上的曲面面積：

按一下以複製此運算式

<region id="ID0EBCWMB" top="960" left="144" height="22.814" width="64.3233333333333" xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math xml:lang="zh_TW">
    <apply xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50">
      <lessOrEqual xml:lang="zh_TW" />
      <apply xml:lang="zh_TW">
        <plus xml:lang="zh_TW" />
        <apply xml:lang="zh_TW">
          <pow xml:lang="zh_TW" />
          <id xml:lang="zh_TW">x</id>
          <real xml:lang="zh_TW">2</real>
        </apply>
        <apply xml:lang="zh_TW">
          <pow xml:lang="zh_TW" />
          <id xml:lang="zh_TW">y</id>
          <real xml:lang="zh_TW">2</real>
        </apply>
      </apply>
      <apply xml:lang="zh_TW">
        <pow xml:lang="zh_TW" />
        <id xml:lang="zh_TW">r</id>
        <real xml:lang="zh_TW">2</real>
      </apply>
    </apply>
    <resultFormat xml:lang="zh_TW">
      <matrix size="12,12" offset="0,0" xml:lang="zh_TW" />
    </resultFormat>
  </math>
</region>

1. 定義含有兩個變數的函數。

按一下以複製此運算式

<region id="ID0E1DWMB" top="1032" left="48" height="22.814" width="110.28" xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math xml:lang="zh_TW">
    <define xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50">
      <function xml:lang="zh_TW">
        <id xml:lang="zh_TW" label="None">f</id>
        <boundVars xml:lang="zh_TW">
          <id xml:lang="zh_TW" label="None">x</id>
          <id xml:lang="zh_TW" label="None">y</id>
        </boundVars>
      </function>
      <apply xml:lang="zh_TW">
        <minus xml:lang="zh_TW" />
        <apply xml:lang="zh_TW">
          <minus xml:lang="zh_TW" />
          <real xml:lang="zh_TW">2</real>
          <apply xml:lang="zh_TW">
            <pow xml:lang="zh_TW" />
            <id xml:lang="zh_TW">x</id>
            <real xml:lang="zh_TW">2</real>
          </apply>
        </apply>
        <apply xml:lang="zh_TW">
          <pow xml:lang="zh_TW" />
          <id xml:lang="zh_TW">y</id>
          <real xml:lang="zh_TW">2</real>
        </apply>
      </apply>
    </define>
    <resultFormat xml:lang="zh_TW">
      <matrix size="12,12" offset="0,0" xml:lang="zh_TW" />
    </resultFormat>
  </math>
</region>

2. 定義圓形積分區域的半徑。

按一下以複製此運算式

3. 計算圓上的曲面面積。

按一下以複製此運算式

<region id="ID0EEHWMB" top="1248" left="48" height="110.551800578833" width="325.3466491355" xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math xml:lang="zh_TW">
    <eval xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50">
      <apply xml:lang="zh_TW">
        <integral xml:lang="zh_TW" />
        <lambda xml:lang="zh_TW">
          <boundVars xml:lang="zh_TW">
            <id xml:lang="zh_TW">x</id>
          </boundVars>
          <apply xml:lang="zh_TW">
            <integral xml:lang="zh_TW" />
            <lambda xml:lang="zh_TW">
              <boundVars xml:lang="zh_TW">
                <id xml:lang="zh_TW">y</id>
              </boundVars>
              <apply xml:lang="zh_TW">
                <nthRoot xml:lang="zh_TW" />
                <placeholder xml:lang="zh_TW" />
                <apply xml:lang="zh_TW">
                  <plus xml:lang="zh_TW" />
                  <apply xml:lang="zh_TW">
                    <plus xml:lang="zh_TW" />
                    <real xml:lang="zh_TW">1</real>
                    <apply xml:lang="zh_TW">
                      <pow xml:lang="zh_TW" />
                      <parens xml:lang="zh_TW">
                        <apply xml:lang="zh_TW">
                          <derivative xml:lang="zh_TW" />
                          <lambda xml:lang="zh_TW">
                            <boundVars xml:lang="zh_TW">
                              <id xml:lang="zh_TW">x</id>
                            </boundVars>
                            <apply xml:lang="zh_TW">
                              <id xml:lang="zh_TW">f</id>
                              <sequence xml:lang="zh_TW">
                                <id xml:lang="zh_TW">x</id>
                                <id xml:lang="zh_TW">y</id>
                              </sequence>
                            </apply>
                          </lambda>
                          <degree xml:lang="zh_TW">
                            <placeholder xml:lang="zh_TW" />
                          </degree>
                        </apply>
                      </parens>
                      <real xml:lang="zh_TW">2</real>
                    </apply>
                  </apply>
                  <apply xml:lang="zh_TW">
                    <pow xml:lang="zh_TW" />
                    <parens xml:lang="zh_TW">
                      <apply xml:lang="zh_TW">
                        <derivative xml:lang="zh_TW" />
                        <lambda xml:lang="zh_TW">
                          <boundVars xml:lang="zh_TW">
                            <id xml:lang="zh_TW">y</id>
                          </boundVars>
                          <apply xml:lang="zh_TW">
                            <id xml:lang="zh_TW">f</id>
                            <sequence xml:lang="zh_TW">
                              <id xml:lang="zh_TW">x</id>
                              <id xml:lang="zh_TW">y</id>
                            </sequence>
                          </apply>
                        </lambda>
                        <degree xml:lang="zh_TW">
                          <placeholder xml:lang="zh_TW" />
                        </degree>
                      </apply>
                    </parens>
                    <real xml:lang="zh_TW">2</real>
                  </apply>
                </apply>
              </apply>
            </lambda>
            <lowerBound xml:lang="zh_TW">
              <apply xml:lang="zh_TW">
                <neg xml:lang="zh_TW" />
                <apply xml:lang="zh_TW">
                  <nthRoot xml:lang="zh_TW" />
                  <placeholder xml:lang="zh_TW" />
                  <apply xml:lang="zh_TW">
                    <minus xml:lang="zh_TW" />
                    <apply xml:lang="zh_TW">
                      <pow xml:lang="zh_TW" />
                      <id xml:lang="zh_TW">r</id>
                      <real xml:lang="zh_TW">2</real>
                    </apply>
                    <apply xml:lang="zh_TW">
                      <pow xml:lang="zh_TW" />
                      <id xml:lang="zh_TW">x</id>
                      <real xml:lang="zh_TW">2</real>
                    </apply>
                  </apply>
                </apply>
              </apply>
            </lowerBound>
            <upperBound xml:lang="zh_TW">
              <apply xml:lang="zh_TW">
                <nthRoot xml:lang="zh_TW" />
                <placeholder xml:lang="zh_TW" />
                <apply xml:lang="zh_TW">
                  <minus xml:lang="zh_TW" />
                  <apply xml:lang="zh_TW">
                    <pow xml:lang="zh_TW" />
                    <id xml:lang="zh_TW">r</id>
                    <real xml:lang="zh_TW">2</real>
                  </apply>
                  <apply xml:lang="zh_TW">
                    <pow xml:lang="zh_TW" />
                    <id xml:lang="zh_TW">x</id>
                    <real xml:lang="zh_TW">2</real>
                  </apply>
                </apply>
              </apply>
            </upperBound>
          </apply>
        </lambda>
        <lowerBound xml:lang="zh_TW">
          <apply xml:lang="zh_TW">
            <neg xml:lang="zh_TW" />
            <id xml:lang="zh_TW">r</id>
          </apply>
        </lowerBound>
        <upperBound xml:lang="zh_TW">
          <id xml:lang="zh_TW">r</id>
        </upperBound>
      </apply>
      <result xml:lang="zh_TW">
        <real xml:lang="zh_TW">13.614</real>
      </result>
    </eval>
    <resultFormat xml:lang="zh_TW">
      <matrix size="12,12" offset="0,0" xml:lang="zh_TW" />
    </resultFormat>
  </math>
</region>

4. 將函數轉換為極座標系統，並計算圓上的曲面面積。

按一下以複製此運算式

按一下以複製此運算式

<region id="ID0ELJWMB" top="1464" left="48" height="99.5330005123615" width="313.209982468834" xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math xml:lang="zh_TW">
    <eval xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50">
      <apply xml:lang="zh_TW">
        <integral xml:lang="zh_TW" />
        <lambda xml:lang="zh_TW">
          <boundVars xml:lang="zh_TW">
            <id xml:lang="zh_TW">θ</id>
          </boundVars>
          <apply xml:lang="zh_TW">
            <integral xml:lang="zh_TW" />
            <lambda xml:lang="zh_TW">
              <boundVars xml:lang="zh_TW">
                <id xml:lang="zh_TW">r</id>
              </boundVars>
              <apply xml:lang="zh_TW">
                <scale xml:lang="zh_TW" />
                <apply xml:lang="zh_TW">
                  <nthRoot xml:lang="zh_TW" />
                  <placeholder xml:lang="zh_TW" />
                  <apply xml:lang="zh_TW">
                    <plus xml:lang="zh_TW" />
                    <apply xml:lang="zh_TW">
                      <plus xml:lang="zh_TW" />
                      <real xml:lang="zh_TW">1</real>
                      <apply xml:lang="zh_TW">
                        <pow xml:lang="zh_TW" />
                        <parens xml:lang="zh_TW">
                          <apply xml:lang="zh_TW">
                            <derivative xml:lang="zh_TW" />
                            <lambda xml:lang="zh_TW">
                              <boundVars xml:lang="zh_TW">
                                <id xml:lang="zh_TW">r</id>
                              </boundVars>
                              <apply xml:lang="zh_TW">
                                <id xml:lang="zh_TW">f</id>
                                <sequence xml:lang="zh_TW">
                                  <id xml:lang="zh_TW">r</id>
                                  <id xml:lang="zh_TW">θ</id>
                                </sequence>
                              </apply>
                            </lambda>
                            <degree xml:lang="zh_TW">
                              <placeholder xml:lang="zh_TW" />
                            </degree>
                          </apply>
                        </parens>
                        <real xml:lang="zh_TW">2</real>
                      </apply>
                    </apply>
                    <apply xml:lang="zh_TW">
                      <pow xml:lang="zh_TW" />
                      <parens xml:lang="zh_TW">
                        <apply xml:lang="zh_TW">
                          <derivative xml:lang="zh_TW" />
                          <lambda xml:lang="zh_TW">
                            <boundVars xml:lang="zh_TW">
                              <id xml:lang="zh_TW">θ</id>
                            </boundVars>
                            <apply xml:lang="zh_TW">
                              <id xml:lang="zh_TW">f</id>
                              <sequence xml:lang="zh_TW">
                                <id xml:lang="zh_TW">r</id>
                                <id xml:lang="zh_TW">θ</id>
                              </sequence>
                            </apply>
                          </lambda>
                          <degree xml:lang="zh_TW">
                            <placeholder xml:lang="zh_TW" />
                          </degree>
                        </apply>
                      </parens>
                      <real xml:lang="zh_TW">2</real>
                    </apply>
                  </apply>
                </apply>
                <id xml:lang="zh_TW">r</id>
              </apply>
            </lambda>
            <lowerBound xml:lang="zh_TW">
              <real xml:lang="zh_TW">0</real>
            </lowerBound>
            <upperBound xml:lang="zh_TW">
              <apply xml:lang="zh_TW">
                <nthRoot xml:lang="zh_TW" />
                <placeholder xml:lang="zh_TW" />
                <real xml:lang="zh_TW">2</real>
              </apply>
            </upperBound>
          </apply>
        </lambda>
        <lowerBound xml:lang="zh_TW">
          <real xml:lang="zh_TW">0</real>
        </lowerBound>
        <upperBound xml:lang="zh_TW">
          <apply xml:lang="zh_TW">
            <scale xml:lang="zh_TW" />
            <real xml:lang="zh_TW">2</real>
            <id xml:lang="zh_TW">π</id>
          </apply>
        </upperBound>
      </apply>
      <result xml:lang="zh_TW">
        <real xml:lang="zh_TW">13.614</real>
      </result>
    </eval>
    <resultFormat xml:lang="zh_TW">
      <matrix size="12,12" offset="0,0" xml:lang="zh_TW" />
    </resultFormat>
  </math>
</region>