学習環境
- Surface 3 (4G LTE)、Surface 3 タイプ カバー、Surface ペン(端末)
- Windows 10 Pro (OS)
- 数式入力ソフト(TeX, MathML): MathType
- MathML対応ブラウザ: Firefox、Safari
- MathML非対応ブラウザ(Internet Explorer, Microsoft Edge, Google Chrome...)用JavaScript Library: MathJax
- 参考書籍
数学読本〈5〉微分法の応用/積分法/積分法の応用/行列と行列式(松坂 和夫(著)、岩波書店)の第20章(面積、体積、長さ - 積分法の応用)、20.1(面積)、面積の公式、問1.を取り組んでみる。
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y=(x+1)(x−1)∫1−1(0−(x2−1))dx=−2∫10x2dx+2∫101dx=−2[13x3]10+2[x]10=−23+2=43
y=(1−x)(x2+x+1)∫10(1−x3)dx=[x−14x4]10=1−14=34
y=x2(1−x)∫10(x2−x3)dx=[13x3−14x4]10=13−14=112
y=−(x2−2x−3)=−(x−3)(x+1)∫3−1(−x2+2x+3)dx=[−13x3+x2+3x]3−1=(−9+9+9)−(13+1−3)=9+53=323
x2=−x2+3x+52x2−3x−5=0(x+1)(2x−5)=0x=−1,52∫20((−x2+3x+5)−x2)dx=∫20(−2x2+3x+5)dx=[−23x3+32x2+5x]20=−163+6+10=323
y=√xx2=√xx4=xx(x3−1)=0x=0,1∫10(√x−x2)dx=[23x32−13x3]10=23−13=13
y2=3xx=y234y−y2=y2312y−3y2=y24y2−12y=04y(y−3)=0y=0,3∫30((4y−y2)−y23)dy=∫30(4y−43y2)dy=[2y2−49y3]30=18−12=6
∫π40cosxdx=[sinx]π40=sinπ4−sin0=1√2
sinx=cosxx=π4∫π40(cosx−sinx)dx=[sinx+cosx]π40=(sinπ4+cosπ4)−(sin0+cos0)=1√2+1√2−(0+1)=2√2−1=√2−1
=∫π0sinxdx−∫2ππsinxdx+∫3π2πsinxdx=2∫π0sinxdx−∫2ππsinxdx=2[−cosx]π0−[−cosx]2ππ=2(−cosπ+cos0)+(cos2π−cosπ)=2(1+1)+(1+1)=6
y=1xy=52−x52−x=1x52x−x2−1=02x2−5x+2=0x=5±√25−164=5±√94=5±34=12,2∫212((52−x)−1x)dx=∫212(52−x−1x)dx=[52x−12x2−logx]212=(5−2−log2)−(54−18−log12)=(3−log2)−(98−log12)=158−log2+log12=158−log2−log2=158−2log2
コード(Emacs)
Python 3
#!/usr/bin/env python3 # -*- coding: utf-8 -*- from sympy import pprint, symbols, Integral, sqrt, cos, pi, sin, Rational print('1.') x = symbols('x') fs = [(0 - (x ** 2 - 1), (-1, 1)), (1 - x ** 3, (0, 1)), (x ** 2 - x ** 3, (0, 1)), (- x ** 2 + 2 * x + 3, (-1, 3)), (-x ** 2 + 3 * x + 5 - x ** 2, (0, 2)), (sqrt(x) - x ** 2, (0, 1)), ((4 * x - x ** 2) - x ** Rational(2, 3), (0, 3)), (cos(x), (0, pi / 4)), (cos(x) - sin(x), (0, pi / 4)), (abs(sin(x)), (0, 3 * pi)), (Rational(5, 2) - x - 1 / x, (Rational(1, 2), 2))] for i, (f, (x1, x2)) in enumerate(fs, 1): print(f'({i})') I = Integral(f, (x, x1, x2)) for g in [I, I.doit()]: pprint(g) print() print()
入出力結果(Terminal, Jupyter(IPython))
$ ./sample1.py 1. (1) 1 ⌠ ⎮ ⎛ 2 ⎞ ⎮ ⎝- x + 1⎠ dx ⌡ -1 4/3 (2) 1 ⌠ ⎮ ⎛ 3 ⎞ ⎮ ⎝- x + 1⎠ dx ⌡ 0 3/4 (3) 1 ⌠ ⎮ ⎛ 3 2⎞ ⎮ ⎝- x + x ⎠ dx ⌡ 0 1/12 (4) 3 ⌠ ⎮ ⎛ 2 ⎞ ⎮ ⎝- x + 2⋅x + 3⎠ dx ⌡ -1 32/3 (5) 2 ⌠ ⎮ ⎛ 2 ⎞ ⎮ ⎝- 2⋅x + 3⋅x + 5⎠ dx ⌡ 0 32/3 (6) 1 ⌠ ⎮ ⎛ 2⎞ ⎮ ⎝√x - x ⎠ dx ⌡ 0 1/3 (7) 3 ⌠ ⎮ ⎛ 2/3 2 ⎞ ⎮ ⎝- x - x + 4⋅x⎠ dx ⌡ 0 2/3 9⋅3 - ────── + 9 5 (8) π ─ 4 ⌠ ⎮ cos(x) dx ⌡ 0 √2 ── 2 (9) π ─ 4 ⌠ ⎮ (-sin(x) + cos(x)) dx ⌡ 0 -1 + √2 (10) 3⋅π ⌠ ⎮ │sin(x)│ dx ⌡ 0 3⋅π ⌠ ⎮ │sin(x)│ dx ⌡ 0 (11) 2 ⌠ ⎮ ⎛ 5 1⎞ ⎮ ⎜-x + ─ - ─⎟ dx ⎮ ⎝ 2 x⎠ ⌡ 1/2 -2⋅log(2) + 15/8 $
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