【微分積分】4-7-1 面積｜問題集

それぞれの関数の\(x\)交点は
\(x^2=x+2\)
\(x=-1,2\)
よって、求める面積\(S\)は
\(\displaystyle S=\int_{-1}^2(x+2-x^2)dx\)
\(\displaystyle =\left[\frac{x^2}{2}+2x-\frac{x^3}{3}\right]_{-1}^2\)
\(\displaystyle =\frac{9}{2}\)

それぞれの関数の\(x\)交点は
\(x^3=x^2\)
\(x=0,1\)
よって、求める面積\(S\)は
\(\displaystyle S=\int_0^1(x^2-x^3)dx\)
\(\displaystyle =\left[\frac{x^3}{3}-\frac{x^4}{4}\right]_0^1\)
\(\displaystyle =\frac{1}{12}\)

\(y=-\sqrt{x},y=x-6\)の\(x\)交点は
\(-\sqrt{x}=x-6\)
\(x=4\)
\(y=x-6,y=0\)の\(x\)交点は
\(x=6\)
よって、求める面積\(S\)は
\(\displaystyle S=\int_0^4\sqrt{x}dx+\int_4^6-(x-6)dx\)
\(\displaystyle =\left[\frac{2}{3}x^{\frac{3}{2}}\right]_0^4-\left[\frac{1}{2}x^2-6x\right]_4^6\)
\(\displaystyle =\frac{22}{3}\)

それぞれの関数の\(x\)交点は
\(x^3-x=1-x^2\)
\(x=\pm1\)
よって、求める面積\(S\)は
\(\displaystyle S=\int_{-1}^1(1-x^2-x^3+x)dx\)
\(\displaystyle =2\int_0^1(-x^2+1)dx\)
\(\displaystyle =2\left[-\frac{x^3}{3}+x\right]_0^1\)
\(\displaystyle =\frac{4}{3}\)

それぞれの関数の\(y\)交点は
\(5+4=y^2\)
\(y=\pm3\)
よって、求める面積\(S\)は
\(\displaystyle S=\int_{-3}^3(5-y^2+4)dy\)
\(\displaystyle =2\int_0^3(9-y^2)dy\)
\(\displaystyle =2\left[9y-\frac{y^3}{3}\right]_0^3\)
\(\displaystyle =36\)

\(y=2x,x+y=9\)の\(x\)交点は
\(x=3\)
\(y=2x,y=x-1\)の\(x\)交点は
\(x=-1\)
\(x+y=9,y=x-1\)の\(x\)交点は
\(x=5\)
よって、求める面積\(S\)は
\(\displaystyle S=\int_{-1}^3(2x-x+1)dx+\int_3^5(9-x-x+1)dx\)
\(\displaystyle =\int_{-1}^3(x+1)dx+\int_3^5(-2x+10)dx\)
\(\displaystyle =\left[\frac{1}{2}x^2+x\right]_{-1}^3+[-x^2+5x]_3^5\)
\(\displaystyle =12\)

それぞれの関数の\(y\)交点は
\(y^2=3-2y^2\)
\(y=\pm1\)
よって、求める面積\(S\)は
\(\displaystyle S=\int_{-1}^1(3-2y^2-y^2)dy\)
\(\displaystyle =\int_{-1}^1(3-3y^2)dy\)
\(\displaystyle =[3y-y^3]_{-1}^1\)
\(\displaystyle =4\)

求める面積\(S\)は
\(\displaystyle S=2\int_0^1y(1-y^{\frac{2}{3}})^{\frac{3}{2}}dy\)
\(\displaystyle =6\int_0^{\frac{\pi}{2}}\sin^5\theta\cos^4\theta d\theta\)
\(\displaystyle =6・\frac{1}{2}B\left(\frac{6}{2},\frac{5}{2}\right)\)
\(\displaystyle =3・\frac{16}{315}\)
\(\displaystyle =\frac{16}{105}\)