Asosiy tarkibga o'tish

6.5-bo'lim: Ko'proq ovoz balandligi muammolari

  1. Bazasi \(L\) uzunlikdagi teng yonli uchburchak bo‘lgan \(h\) balandlikdagi piramidaning hajmini toping. Yechim
  2. Bazasi \(r\) radiusli disk va kesmalari kvadrat bo‘lgan qattiq jismning hajmini toping. Ko'ndalang kesimlarning eskizini ko'rish uchun quyidagi rasmga qarang.
    A sketch showing a typical cross-section of the solid. The base of the solid is the disk \({x^2} + {y^2} = {r^2}\), drawn in perspective as an ellipse, and a shaded square stands vertically on the disk with its base running across the disk. The distance from the center of the disk out along the base is labeled \(x\) and half the width of the square is labeled \(y\).
    Yechim
  3. Bazasi \(x = 2 - {y^2}\) va \(x = {y^2} - 2\) bilan chegaralangan mintaqa bo‘lgan va ko‘ndalang kesimlari asosi \(y\) o‘qiga perpendikulyar bo‘lgan teng yonli uchburchaklar va asosi va uzunligi teng bo‘lgan ikki tomoni orasidagi burchak \(\frac{\pi }{4}\) bo‘lgan qattiq jismning hajmini toping. Ko'ndalang kesimlarning eskizini ko'rish uchun quyidagi rasmga qarang.
    A sketch showing a typical cross-section of the solid. The base of the solid is the region between the two parabolas \(x = {y^2} - 2\) and \(x = 2 - {y^2}\), drawn in perspective, and a shaded isosceles triangle stands vertically on the base with its base running across the region. The two base angles of the triangle are each labeled \(\frac{\pi }{4}\), the distance out along the base is labeled \(x\) and half the width is labeled \(y\).
    Yechim
  4. Bazasi \(y = \sqrt {4 - x} \), \(x = - 4\) va \(x\) o‘qi bilan chegaralangan hudud bo‘lgan “silindr”dan kesilgan xanjar hajmini toping. Takozning yuqori va pastki qismlari orasidagi burchak \(\frac{\pi }{3}\). "Tsilindr" va takozning eskizini quyidagi rasmga qarang (eskizda ijobiy \(x\) o'qi va musbat \(y\) o'qi ko'rsatilgan - ular faqat boshqa yo'nalishda).
    Two sketches side by side. The one on the left, labeled Cylinder, shows the solid whose base is the region bounded by \(y = \sqrt {4 - x} \), \(x = - 4\) and the x-axis, drawn as a tall vertical column standing on that base. The one on the right, labeled Wedge, shows the thin wedge cut out of that column by a plane through the x-axis making an angle of \(\frac{\pi }{3}\) with the base. In both sketches the curve \(y = \sqrt {4 - x} \) and the line \(x = - 4\) are labeled and the angle \(\frac{\pi }{3}\) is marked at the near corner.
    Yechim