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重積分の計算問題

次の重積分を計算せよ．
- (*) $3$\displaystyle{\int _{0}^{2} \int _{ \frac{\hspace{2}y\hspace{2}}{\hspace{2}2\hspace{2}} }^{1} \( x+y \) dxdy }$ ⇒ 解答
- (*) $3$\displaystyle{\int _{0}^{1} \int _{0}^{ 1- x } \( {x}^{2}- 2y \) dydx }$ ⇒ 解答
- (*) $3$\displaystyle{\int _{0}^{{\pi}} \int _{0}^{{\pi}} \sin \( x+y \) dxdy }$ ⇒ 解答
次の重積分を計算せよ．
1. $3$\displaystyle{ \displaystyle{ \hspace{1}{\int\int }_{D} ydxdy } }$ $3$\displaystyle{\(D:0\leq x\leq 1,0\leq y\leq {x}^{2}\)}$ ⇒ 解答
2. $3$\displaystyle{ \displaystyle{ \hspace{1}{\int\int }_{D} \( {x}^{3}+xy \) dxdy } }$ $3$\displaystyle{\(D:0\leq y\leq 1,0\leq x\leq \sqrt{y}\)}$ ⇒ 解答
3. $3$\displaystyle{ \displaystyle{ \hspace{1}{\int\int }_{D} \( {x}^{2}- 2y \) dxdy } }$ $3$\displaystyle{\(D:x+y\leq 1,x\geq 0,y\geq 0\)}$ ⇒ 解答　
4. $3$ \displaystyle{ \hspace{1}{\int\int }_{D} \log xy }dxdy$ $3$\displaystyle{ \( D:1\leq x\leq 2,1\leq y\leq 2 \) }$ ⇒ 解答
5. $3$\displaystyle{ \hspace{1}{\int\int }_{D} \( 2x- y \) dxdy }$ $3$\displaystyle{ \( \text{\hspace{1} }D\text{\hspace{1} }:\text{\hspace{1} }\text{\hspace{1} }- 3\leq x\leq 3\text{\hspace{1} },\text{\hspace{1} }0\leq y\leq 3\text{\hspace{1} } \) }$ ⇒ 解答
6. $3$\displaystyle{ \hspace{1}{\int\int }_{D} x{y}^{2}dxdy }$ $3$\displaystyle{ \( \text{\hspace{1} }D\text{\hspace{1} }:\text{\hspace{1} }0\leq x\leq 2\text{\hspace{1} },\text{\hspace{1} }- 2\leq y\leq 1\text{\hspace{1} } \) }$ ⇒ 解答
7. $3$\displaystyle{ \hspace{1}{\int\int }_{D} \( x- 2xy+3y \) dxdy }$ $3$\displaystyle{ \( \text{\hspace{1} }D\text{\hspace{1} }:\text{\hspace{1} }- 1\leq x\leq 1\text{\hspace{1} },\text{\hspace{1} }- 2\leq y\leq 0\text{\hspace{1} } \) }$ ⇒ 解答
8. $3$\displaystyle{ \hspace{1}{\int\int }_{D} \( {x}^{2}+{y}^{2} \) dxdy }$ $3$\displaystyle{ \( \text{\hspace{1} }D\text{\hspace{1} }:\text{\hspace{1} }- 2\leq x\leq 2\text{\hspace{1} },\text{\hspace{1} }- 2\leq y\leq 2\text{\hspace{1} } \) }$ ⇒ 解答
9. $3$\displaystyle{ \hspace{1}{\int\int }_{D} \sin \( x+y \) dxdy }$ $3$\displaystyle{ \( \text{\hspace{1} }D\text{\hspace{1} }:\text{\hspace{1} }0\leq x\leq {\pi}\text{\hspace{1} },\text{\hspace{1} }0\leq y\leq {\pi}\text{\hspace{1} } \) }$ ⇒ 解答
10. $3$\displaystyle{ \hspace{1}{\int\int }_{D} 2x\cos ydxdy }$ $3$\displaystyle{ \( \text{\hspace{1} }D\text{\hspace{1} }:\text{\hspace{1} }0\leq x\leq \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}}\text{\hspace{1} },\text{\hspace{1} }\frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}6\hspace{2}}\leq y\leq \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}}\text{\hspace{1} } \) }$ ⇒ 解答
11. $3$\displaystyle{ \hspace{1}{\int\int }_{D} r\sin {\theta}drd{\theta} }$ $3$\displaystyle{ \( \text{\hspace{1} }D\text{\hspace{1} }:\text{\hspace{1} }\frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}3\hspace{2}}\leq {\theta}\leq {\pi}\text{\hspace{1} },\text{\hspace{1} }0\leq r\leq 4\text{\hspace{1} } \) }$ ⇒ 解答
12. $3$\displaystyle{ \hspace{1}{\int\int }_{D} xdxdy }$ $3$\displaystyle{ \( \text{\hspace{1} }D\text{\hspace{1} }:\text{\hspace{1} }\frac{\hspace{2}x\hspace{2}}{\hspace{2}2\hspace{2}}+\frac{\hspace{2}y\hspace{2}}{\hspace{2}4\hspace{2}}\leq 1\text{\hspace{1} },\text{\hspace{1} }x\geq 0\text{\hspace{1} },\text{\hspace{1} }y\geq 0\text{\hspace{1} } \) }$ ⇒ 解答
13. $3$\displaystyle{ \hspace{1}{\int\int }_{D} \( x+y \) dxdy }$ $3$\displaystyle{ \( \text{\hspace{1} }D\text{\hspace{1} }:\text{\hspace{1} }\frac{\hspace{2}x\hspace{2}}{\hspace{2}3\hspace{2}}+y\leq 2\text{\hspace{1} },\text{\hspace{1} }x\geq 0\text{\hspace{1} },\text{\hspace{1} }y\geq 0\text{\hspace{1} } \) }$ ⇒ 解答
14. $3$\displaystyle{ \hspace{1}{\int\int }_{D} xydxdy }$ $3$\displaystyle{ \( \text{\hspace{1} }D\text{\hspace{1} }:\text{\hspace{1} }2x+\frac{\hspace{2}y\hspace{2}}{\hspace{2}2\hspace{2}}\leq 1\text{\hspace{1} },\text{\hspace{1} }x\geq 0\text{\hspace{1} },\text{\hspace{1} }y\geq 0\text{\hspace{1} } \) }$ ⇒ 解答
15. $3$\displaystyle{ \hspace{1}{\int\int }_{D} \( {x}^{2}+{y}^{2} \) dxdy }$ $3$\displaystyle{ \( \text{\hspace{1} }D\text{\hspace{1} }:\text{\hspace{1} }x+y\leq 2\text{\hspace{1} },\text{\hspace{1} }x\geq 0\text{\hspace{1} },\text{\hspace{1} }y\geq 0\text{\hspace{1} } \) }$ ⇒ 解答
16. $3$\displaystyle{ \hspace{1}{\int\int }_{D} \sin \( x+y \) dxdy }$ $3$\displaystyle{ \( \text{\hspace{1} }D\text{\hspace{1} }:\text{\hspace{1} }x+y\leq {\pi}\text{\hspace{1} },\text{\hspace{1} }x\geq 0\text{\hspace{1} },\text{\hspace{1} }y\geq 0\text{\hspace{1} } \) }$ ⇒ 解答
17. $3$\displaystyle{ \hspace{1}{\int\int }_{D} \( 4- x- y \) dxdy }$ $3$\displaystyle{ \( \text{\hspace{1} }D\text{\hspace{1} }:\text{\hspace{1} }0\leq x\leq 2\text{\hspace{1} },\text{\hspace{1} }- x\leq y\leq x\text{\hspace{1} } \) }$ ⇒ 解答
18. $3$\displaystyle{ \hspace{1}{\int\int }_{D} { \( x- y \) }^{2}dxdy }$ $3$\displaystyle{ \( \text{\hspace{1} }D\text{\hspace{1} }:\text{\hspace{1} } \| x+2 \| \leq 1\text{\hspace{1} },\text{\hspace{1} } \| x- 2y \| \leq 1\text{\hspace{1} } \) }$ ⇒ 解答
19. $3$\displaystyle{ \hspace{1}{\int\int }_{D} xdxdy }$ $3$\displaystyle{ \( \text{\hspace{1} }D\text{\hspace{1} }:\text{\hspace{1} }0\leq y\leq 3x- {x}^{2}\text{\hspace{1} } \) }$ ⇒ 解答
20. $3$\displaystyle{ \hspace{1}{\int\int }_{D} \( x+2 \) dxdy }$ $3$\displaystyle{ \( \text{\hspace{1} }D\text{\hspace{1} }:\text{\hspace{1} }- 1\leq x\leq 1\text{\hspace{1} },\text{\hspace{1} }0\leq y\leq {x}^{2}+1\text{\hspace{1} } \) }$ ⇒ 解答
21. $3$\displaystyle{ \hspace{1}{\int\int }_{D} \( 2xy+3{y}^{2} \) dxdy }$ $3$\displaystyle{ \( \text{\hspace{1} }D\text{\hspace{1} }:\text{\hspace{1} }{x}^{2}\leq y\leq x\text{\hspace{1} } \) }$ ⇒ 解答
22. $3$\displaystyle{ \hspace{1}{\int\int }_{D} \( 2x+y \) dxdy }$ $3$\displaystyle{ \( \text{\hspace{1} }D\text{\hspace{1} }:\text{\hspace{1} }{x}^{2}\leq y\leq x+2\text{\hspace{1} } \) }$ ⇒ 解答
23. $3$\displaystyle{ \hspace{1}{\int\int }_{D} xydxdy }$ $3$\displaystyle{ \( D:{x}^{2}\leq y\leq x \) }$ ⇒ 解答
24. $3$\displaystyle{ \hspace{1}{\int\int }_{D} {y}^{2}dxdy }$ $3$\displaystyle{ \( D:{x}^{2}+{y}^{2}\leq 9,x\geq 0 \) }$ ⇒ 解答
25. $3$\displaystyle{\displaystyle{ \hspace{1}{\int\int }_{D} \( {x}^{2}- {y}^{2} \) } \( x+y \) dxdy}$ $3$\displaystyle{ \( D:0\leq x+y\leq 2,- 1\leq x- y\leq 2 \) }$ ⇒ 解答
26. $3$\displaystyle{ \displaystyle{ \hspace{1}{\int\int }_{D} 4 \( {x}^{2}+2xy+{y}^{2} \) } \( {x}^{2}- {y}^{2} \) dxdy }$ $3$\displaystyle{ \( D:0\leq x+y\leq 2,- 1\leq x- y\leq 2 \) }$ ⇒ 解答
27. $3$\displaystyle{\displaystyle{ \hspace{1}{\int\int }_{D} x\sin xydxdy }}$ $3$\displaystyle{ \( D:0\leq x\leq 1,0\leq y\leq \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}} \) }$ ⇒ 解答
28. $3$\displaystyle{\displaystyle{ \hspace{1}{\int\int }_{D} {e}^{x}\cos ydxdy }}$ $3$\displaystyle{ \( D:0\leq x\leq 1,0\leq y\leq \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}4\hspace{2}} \) }$ ⇒ 解答
29. $3$\displaystyle{ \displaystyle{ \hspace{1}{\int\int }_{D} {e}^{x}\sin ydxdy } }$ $3$\displaystyle{ \( D:0\leq x\leq 1,0\leq y\leq {\pi}x \) }$ ⇒ 解答
30. $3$\displaystyle{\displaystyle{ \hspace{1}{\int\int }_{D} \frac{\hspace{2} x- y \hspace{2}}{\hspace{2} 1+x+y \hspace{2}} }dxdy}$ $3$\displaystyle{ \( D:0\leq x+y\leq 1,0\leq x- y\leq 1 \) }$ ⇒ 解答
31. $3$\displaystyle{\displaystyle{ \hspace{1}{\int\int }_{D} 64xydxdy }}$ $3$\displaystyle{ \( D:1\leq x+y\leq 2,- 2\leq x- 3y\leq 0 \) }$ ⇒ 解答
32. $3$\displaystyle{\displaystyle{ \hspace{1}{\int\int }_{D} \left({ 2x- y }\right)dx }dy}$ $3$\displaystyle{ \left({ D:0≦x≦1,x≦y≦\sqrt{x} }\right) }$
  TeXに変換設定していない数学記号や，特殊文字が含まれています。今後直していきます。
  ⇒ 解答
33. $3$\displaystyle{\displaystyle{ \hspace{1}{\int\int }_{D} \sqrt{ {a}^{2}- {x}^{2}- {y}^{2} }dxdy }}$ $3$\displaystyle{ \(D:{x}^{2}+{y}^{2}\leq ax\) }$ ⇒ 解答
34. $3$\displaystyle{ \displaystyle{ \hspace{1}{\int\int }_{D} \( {x}^{2}+{y}^{2} \) {e}^{ - x- y } }dxdy }$ $3$\displaystyle{ \(D:- 2< x+y< 2,- 2< x- y< 2\) }$ ⇒ 解答
35. $3$\displaystyle{\displaystyle{ \hspace{1}{\int\int }_{D} 4 \( {x}^{2}+2xy+{y}^{2} \) } \( {x}^{2}- {y}^{2} \) dxdy}$ $3$\displaystyle{ \( D:0\leq x+y\leq 2,- 1\leq x- y\leq 2 \) }$ ⇒ 解答
36. $3$\displaystyle{ \displaystyle{\hspace{1}{\int\int }_{D} \left({ {x}^{3}+ {y}^{3} }\right) }dxdy }$ $3$\displaystyle{ \( D:0\leq x+y\leq 1,- 1\leq x- 2y\leq 2 \) }$ ⇒ 解答
次の問題に答えよ．
1. 次の変数変換の場合のヤコビアン $3$\displaystyle{ J \( u,v \) }$ を計算せよ．
  $3$\displaystyle{ x=6u- 7v }$ ， $3$\displaystyle{y=3u+5v}$ ⇒ 解答
2. 積分の順序を変更して次の重積分を求めよ．
  $3$\displaystyle{\displaystyle{\int }_{0}^{ 2a }\left({ \displaystyle{\int }_{ \frac{\hspace{2} {x}^{2} \hspace{2}}{\hspace{2}a\hspace{2}} }^{ 6a- x }ydy }\right)dx}$ ⇒ 解答

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最終更新日： 2025年8月21日