1.3: Площині
Плоскі ділянки, в яких рівняння задано вx−y координатах
У нас криваy=y(x) (Figure I.3) and we wish to find the position of the centroid of the area under the curve between x=a and x=b. We consider an elemental slice of width δx на відстані,x from the y axis. Its area is yδx, і тому загальна площа
A=∫baydx
Перший момент площі зрізу по відношенню доy axis is xyδx, and so the first moment of the entire area is ∫baxydx.
Тому
¯x=∫baxydyx∫baydyx=∫baxydyxAlabeleq:1.3.2
Бо¯y ми помічаємо, що відстань центроїда зрізу від xосі є 12y, отже, перший момент області про x вісь є12y.yδx .
Тому
¯y=∫bay2dx2A
Розглянемо напівкруглу пластинку,x2+y2=a2 , see Figure I.4:
Ми маємо справу з частинами як вище, так\(x \) axis, so the area of the semicircle is 2∫a0ydx і нижче і перший момент області є2∫a0xydx.
Ви повинні знайти¯x=4a/(3π)=0.4244a.
Тепер розглянемо ламінатx2+y2=a2 , y>0 (Figure I.5):
Площа елементарного зрізу на цей разyδx (not 2yδx ), and the integration limits are from −a to +a. To find ¯y, use Equation ???, and you should get y=0.4244a .
Plane areas in which the equation is given in polar coordinates.
We consider an elemental triangular sector (Figure I.6) between θ and θ+δθ . The "height" of the triangle is r and the "base" is rδθ. The area of the triangle is 12r2δθ.
Therefore the whole area =
12∫βαr2dθ
The horizontal distance of the centroid of the elemental sector from the origin (more correctly, from the "pole" of the polar coordinate system) is 23rcosθ . The first moment of area of the sector with respect to the y axis is
23rcosθ×12r2δθ=13r3cosθδθ
so the first moment of area of the entire figure between θ=α and θ=β is
13∫βαr3cosθdθ
Therefore
¯x=2∫βαr3cosθdθ3∫βαr2dθ
Similarly
¯x=2∫βαr3sinθdθ3∫βαr2dθ
Consider the semicircle r=a, θ=−π2 to +π2
¯x=2a∫+π/2−π/2cosθdθ3∫+π/2−π/2dθ=2a3π∫+π/2−π/2cosθdθ=4a3π
The reader should now try to find the position of the centroid of a circular sector (slice of pizza!) of angle 2α . The integration limits will be −α to +α.
When you arrive at a formula (which you should keep in a notebook for future reference), check that it goes to 4α3π if α=π2, and to 2π3 if α=0.