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15.3-bo'lim : Umumiy hududlar bo'yicha ikki karrali integrallar

  1. \( \displaystyle \iint\limits_{D}{{8y{x^3}\,dA}}\) ni baholang, bu erda \(D = \left\{ {\left( {x,y} \right)| - 1 \le y \le 2, - 1 \le x \le 1 + {y^2}} \right\}\)
  2. \(D = \left\{ {\left( {x,y} \right)| - 2 \le x \le 2, - {x^2} \le y \le {x^2}} \right\}\) ni \( \displaystyle \iint\limits_{D}{{12{x^2}y - {y^2}\,dA}}\) ni baholang
  3. \( \displaystyle \iint\limits_{D}{{9 - \frac{{6{y^2}}}{{{x^2}}}\,dA}}\) ni baholang, bu erda \(D\) 1-dagi mintaqadirst \(y = {x^3}\) va \(y = 4x\) bilan chegaralangan kvadrant.
  4. \( \displaystyle \iint\limits_{D}{{15{x^2} - 6y\,dA}}\) ni baholang, bu erda \(D\) \(x = \frac{1}{2}{y^2}\) va \(x = 4\sqrt y \) bilan chegaralangan mintaqadir.
  5. \( \displaystyle \iint\limits_{D}{{6y{{\left( {x + 6} \right)}^2}\,dA}}\) ni baholang, bu erda \(D\) \(x = - {y^2}\) va \(x = y - 6\) bilan chegaralangan hududdir.
  6. \( \displaystyle \iint\limits_{D}{{{{\bf{e}}^{{y^{\,2}} + 1}}\,dA}}\) ni baholang, bunda \(D\) uchburchakdir \(\left( {0,0} \right)\), \(\left( { - 2,4} \right)\) va \(\left( {8,4} \right)\).
  7. \( \displaystyle \iint\limits_{D}{{7{y^3}{{\bf{e}}^{{x^{\,2}} + 1}}\,dA}}\) ni baholang, bu erda \(D\) \(y = 2\,\,\sqrt[4]{x}\), \(x = 9\) va \(x\) o'qi bilan chegaralangan hududdir.
  8. \( \displaystyle \iint\limits_{D}{{{x^5}\sin \left( {{y^4}} \right)\,dA}}\) ni baholang, bu erda \(D\) 2-dagi mintaqadir.nd kvadrant \(y = 3{x^2}\), \(y = 12\) va \(y\) o'qi bilan chegaralangan.
  9. \( \displaystyle \iint\limits_{D}{{xy - {y^2}\,dA}}\) ni baholang, bu erda \(D\) quyida ko'rsatilgan mintaqadir.
    This region consists of two triangles. The first triangle has vertices (0,0), (2,4) and (2,-4). The second has vertices at (0,0), (-4,4) and (-4,-4).
  10. \( \displaystyle \iint\limits_{D}{{12{x^3} - 3\,dA}}\) ni baholang, bu erda \(D\) quyida ko'rsatilgan mintaqadir.
    This region consists of two sub regions. The first sub region is the region in the fourth quadrant bounded by $y=x-1$, the x-axis and the y-axis. The second sub region is the region in the second quadrant bounded by $y=1-x^{2}$, the x-axis and the y-axis.
  11. \( \displaystyle \iint\limits_{D}{{6{y^2} + 10y{x^4}\,dA}}\) ni baholang, bu erda \(D\) quyida ko'rsatilgan mintaqadir.
    This region consists of two sub regions. The first sub region is the region in the first and fourth quadrants bounded by $x=2-y^{2}$ and the y-axis. The second sub region is the region in the second and third quadrants bounded by $x=y^{2}-1$ and the y-axis.
  12. \( \displaystyle \iint\limits_{D}{{\frac{{{x^3}}}{{{y^2}}}\,dA}}\) ni baholang, bunda \(D\) \(\displaystyle y = \frac{1}{{{x^2}}}\), \(x = 1\) va \(\displaystyle y = \frac{1}{4}\) bilan chegaralangan hudud bo‘lib, quyida keltirilgan tartibda.
    1. Avval \(x\), keyin esa \(y\) ga nisbatan integratsiya qiling.
    2. Avval \(y\), keyin esa \(x\) ga nisbatan integratsiya qiling.
  13. \( \displaystyle \iint\limits_{D}{{xy - {y^3}\,dA}}\) ni baholang, bunda \(D\) \(y = {x^2}\), \(y = - {x^2}\) va \(x = 2\) bilan chegaralangan hudud bo‘lib, quyida keltirilgan tartibda.
    1. Avval \(x\), keyin esa \(y\) ga nisbatan integratsiya qiling.
    2. Avval \(y\), keyin esa \(x\) ga nisbatan integratsiya qiling.

14-16-masalalar uchun avvalo integrallash tartibini teskari qilib, berilgan integralni baholang.

  1. \(\displaystyle \int_{0}^{8}{{\int_{{{y^{\frac{1}{3}}}}}^{2}{{\frac{y}{{{x^7} + 1}}\,dx}}\,dy}}\)
  2. \(\displaystyle \int_{{ - 4}}^{0}{{\int_{{\sqrt { - x} }}^{2}{{{x^{ - \,\,\frac{2}{3}}}\,\,\sqrt {{y^{\frac{5}{3}}} + 1} \,dy}}\,dx}}\)
  3. \(\displaystyle \int_{0}^{2}{{\int_{{ - x}}^{{3x}}{{5{y^2}{x^3} + 2\,dy}}\,dx}}\)
  4. \(x = - {y^2}\) va \(x = y - 6\) bilan chegaralangan mintaqa maydonini aniqlash uchun ikki karrali integraldan foydalaning.
  5. \(y = {x^2} + 1\) va \(\displaystyle y = \frac{1}{2}{x^2} + 3\) bilan chegaralangan mintaqa maydonini aniqlash uchun ikki karrali integraldan foydalaning.
  6. \(xy\)-tekisligi va \(f\left( {x,y} \right) = 2 - x{y^2}\) orasidagi va \(xy\)-tekisligida \(y = {x^2}\) va \(x = 1\) bilan chegaralangan mintaqadan yuqorida joylashgan hudud hajmini aniqlash uchun qoʻsh integraldan foydalaning.
  7. \(xy\)-tekisligi va \(f\left( {x,y} \right) = 1 + {y^5}\,\,\sqrt {{x^4} + 1} \) orasidagi va \(y = \sqrt x \), \(x = 2\) va \(x\) o'qi bilan chegaralangan \(xy\) tekisligidagi mintaqadan yuqorida joylashgan hudud hajmini aniqlash uchun ikki karrali integraldan foydalaning.
  8. \(2x + 6y + 4z = 8\) tomonidan berilgan tekislikdan pastda joylashgan birinchi oktantdagi mintaqa hajmini aniqlash uchun qoʻsh integraldan foydalaning.
  9. \(z = 3 - 2y\), \(y = 1 - {x^2}\) yuzasi va \(y = 0\) va \(z = 0\) tekisliklari bilan chegaralangan mintaqa hajmini aniqlash uchun ikki karrali integraldan foydalaning.
  10. \(z = 4 - 2x - 2y\), \(y = 2x\), \(y = 0\) va \(z = 0\) tekisliklari bilan chegaralangan mintaqa hajmini aniqlash uchun ikki karrali integraldan foydalaning.
  11. Asos, \(b\) va balandligi \(h\) bo'lgan to'g'ri burchakli uchburchakning maydoni formulasini aniqlash uchun ikki karrali integraldan foydalaning.
  12. Quyidagi rasm uchun formulani aniqlash uchun ikki karrali integraldan foydalaning.
    A sketch of the figure described in the problem. It is a shaded four sided figure with a horizontal base of length \(b\), a vertical left side of height \({h_1}\), a vertical right side of height \({h_2}\) that is shorter than the left side, and a straight slanted top joining the tops of the two sides.