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2曲線と面積の問題の解法 | 夢を叶える塾
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2曲線と面積の問題の解法

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2曲線に囲まれた図形の面積を求める問題です。

基本問題

次の2曲線に囲まれた図形の面積Sを求めなさい。
\displaystyle y=-x^2+\frac{14}{3}x-\frac{5}{3}
\displaystyle y=\frac{1}{2}x^2-\frac{17}{6}x+\frac{13}{3}

解き方

関数を積分する事で関数が描く曲線とx軸の作る面積を求める事が出来ました。
面積の問題は

  1. 覆うものから切り取る
  2. いくつかに分けて足し合わせる

いずれかの方法を使って求める事が多くあります。
今回は2曲線です。
2曲線の場合はどのようにして求めたら良さそうでしょうか?
まずは図を描くところから始めましょう。

解説

次の2曲線に囲まれた図形の面積Sを求めなさい。
\displaystyle y=-x^2+\frac{14}{3}x-\frac{5}{3}
\displaystyle y=\frac{1}{2}x^2-\frac{17}{6}x+\frac{13}{3}

まずは図を描きます。
それぞれ頂点位を求めて、上に凸、下に凸を表現すれば良いです。
※後述しますがグラフの位置関係さえつかめれば、頂点まで求める必要はありません。

\displaystyle y=-x^2+\frac{14}{3}x-\frac{5}{3}
\displaystyle y=-(x-\frac{7}{3})^2+\left(\frac{7}{3}\right)^2-\frac{5}{3}
\displaystyle y=-(x-\frac{7}{3})^2+\frac{49}{9}-\frac{15}{9}
\displaystyle y=-(x-\frac{7}{3})^2+\frac{34}{9}
頂点は\displaystyle (\frac{7}{3},\frac{34}{9})で上に凸です。

\displaystyle y=\frac{1}{2}x^2-\frac{17}{6}x+\frac{13}{3}
\displaystyle y=\frac{1}{2}(x-\frac{17}{6})^2-\frac{1}{2}\left(\frac{17}{6}\right)^2+\frac{13}{3}
\displaystyle y=\frac{1}{2}(x-\frac{17}{6})^2-\frac{289}{72}+\frac{312}{72}
\displaystyle y=\frac{1}{2}(x-\frac{17}{6})^2+\frac{23}{72}
頂点は\displaystyle (\frac{17}{6},\frac{23}{72})で下に凸です。

2曲線の面積_1

こんな感じになりますね。
ここまできれいじゃなくても良いですが、グラフの位置関係はしっかりつかめるように書きましょう。

この斜線部の面積ですがどう求めたらよいでしょうか?

  1. 覆うものから切り取る
  2. いくつかに分けて足し合わせる

ちなみにそれぞれの関数とx軸が作る面積は
2曲線の面積_2

2曲線の面積_3

の様になりますね。

覆うもの\displaystyle y=-x^2+\frac{14}{3}x-\frac{5}{3}から、\displaystyle y=\frac{1}{2}x^2-\frac{17}{6}x+\frac{13}{3}を切り取れば良さそうですね?
2曲線と面積の問題は、いずれかからいずれかを引くことになります。
2曲線を積分する事になります。
そしてその積分区間は2曲線の交点から交点までの区間です。
このことを踏まえると、上記の様に頂点を求めなくても絵は描けますね。
交点をプロットし、その間を上をいく曲線と下を行く曲線を描けば計算できます。

  1. 交点を求める
  2. どちらの曲線が上にあるか(yが大きいか)
  3. 上にある曲線を交点から交点までの区間で積分したものから、下にある曲線のそれを引く式を作る
  4. 積分の計算をする

やってみましょう。

交点を求める
まず交点を求めるためには、2曲線の方程式で作る連立方程式の解を求めます。
\displaystyle y=-x^2+\frac{14}{3}x-\frac{5}{3}
\displaystyle y=\frac{1}{2}x^2-\frac{17}{6}x+\frac{13}{3}
代入法でやってみます。
\displaystyle -x^2+\frac{14}{3}x-\frac{5}{3}=\frac{1}{2}x^2-\frac{17}{6}x+\frac{13}{3}
\displaystyle -6x^2+28x-10=3x^2-17x+26
\displaystyle 9x^2-45x+36=0
\displaystyle x^2-5x+4=0
\displaystyle (x-1)(x-4)=0
という事でx=1,4です。
このときyはy=2,1です。
交点は(1,2),(4,1)になります。

どちらの曲線が上にあるか(yが大きいか)
上に凸のグラフが上に来ますね。
\displaystyle y=-x^2+\frac{14}{3}x-\frac{5}{3}が上です。
仮に\displaystyle g(x)=-x^2+\frac{14}{3}x-\frac{5}{3}としておきましょう。
\displaystyle y=\frac{1}{2}x^2-\frac{17}{6}x+\frac{13}{3}が下です。
仮に\displaystyle h(x)=\frac{1}{2}x^2-\frac{17}{6}x+\frac{13}{3}としておきましょう。

上にある曲線を交点から交点までの区間で積分したものから、下にある曲線のそれを引く式を作る
覆う面積が上にある曲線の面積ですから
\displaystyle \int_1^4 g(x) dx
引く面積が下にある曲線の面積ですから
\displaystyle \int_1^4 h(x) dx

よって面積Sは、
\displaystyle S=\int_1^4 g(x) dx-\int_1^4 h(x) dx
ですね。

積分の計算をする
\displaystyle S=\int_1^4 g(x) dx-\int_1^4 h(x) dx
ですが積分区間が同じなので、併せてしまった方が楽かもしれません
\displaystyle S=\int_1^4 g(x)-h(x) dx
計算していきます。

\displaystyle S=\int_1^4 \left(-x^2+\frac{14}{3}x-\frac{5}{3}\right)-\left(\frac{1}{2}x^2-\frac{17}{6}x+\frac{13}{3}\right) dx
ちなみにこの計算は交点を求めるときに一度計算していますね。
符号とかは気を付けてくださいね。
\displaystyle S=\int_1^4 -\frac{1}{6}(9x^2-45x+36) dx
\displaystyle =-\frac{1}{6}\left[3x^3-\frac{45}{2}x^2+36x\right]_1^4
\displaystyle =-\frac{1}{6}\left(\left(3\cdot 4^3-\frac{45}{2}4^2+36\cdot 4\right)-\left(3-\frac{45}{2}+36\right)\right)
\displaystyle =-\frac{1}{6}\left( 192-360+144-3+\frac{45}{2}-36\right)
\displaystyle =-\frac{1}{6}\left( -63+\frac{45}{2}\right)
\displaystyle =-\frac{1}{12}\left( -126+45\right)
\displaystyle =-\frac{1}{12}\left( -81\right)
\displaystyle =\frac{27}{4}
答えは\displaystyle S=\frac{27}{4}になります。

ちなみに\displaystyle \frac{1}{6}公式なる有名な公式がありますね。
\displaystyle \int_a^b (x-a)(x-b) dx = -\frac{1}{6}(b-a)^3

これに当てはめると
\displaystyle S=\int_1^4 -\frac{1}{6}(9x^2-45x+36) dx
\displaystyle =\int_1^4 -\frac{9}{6}(x-1)(x-4) dx
\displaystyle = -\frac{9}{6}\cdot -\frac{1}{6}(4-1)^3
\displaystyle = \frac{3}{2}\cdot \frac{1}{6}3^3
\displaystyle = \frac{27}{4}
交点が解なので因数分解したときちょうど使える形になるんですよね。

使わなくても答えは出せますので、無理に覚えなくても大丈夫です。
公式を当てはめ間違えて失点するよりは良いかもしれません。
ただ、時間は少し節約できるかもしれませんからね。

終わりに

面積を求める問題は、まずは図を描くところから始めましょう。
どこからどこを引けばいいか等がわかりやすくなります。

しかし色々な図形の面積を積分で求める事ができるようになりますね。
半円の面積を積分で求めたくなったりしますね?
でも数Ⅱの積分の知識だと√の積分ができないですね。
これは数Ⅲでできるようになります。
ワクワクしてきますね。

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