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Calculus: Arc Length, Surface Area, Work, Fluid Forces, and Center of Mass

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이 집합의 용어 (20)

  • Arc length formula for y = f(x)
    If f' is continuous on [a, b], the length of the curve y = f(x) is \(L=\int_a^b \sqrt{1+(f'(x))^2} \, dx\).
  • Arc length formula for x = g(y)
    If g' is continuous on [c, d], the length of the curve x = g(y) is \(L=\int_c^d \sqrt{1+(g'(y))^2} \, dy\).
  • Surface area of revolution about x-axis
    For y = f(x), a ≤ x ≤ b, surface area is \(S=2\pi \int_a^b f(x) \sqrt{1+(f'(x))^2} \, dx\).
  • Surface area of revolution about y-axis
    For x = g(y), c ≤ y ≤ d, surface area is \(S=2\pi \int_c^d g(y) \sqrt{1+(g'(y))^2} \, dy\).
  • Work done lifting a liquid slab
    Work = force × distance, where force is the weight of the slab and distance is how far it is lifted; integrate over the height of the liquid.
  • Fluid force on a submerged vertical plate
    Total force is \(F=\int_a^b \rho g (h - y) L(y) \, dy\), where L(y) is the width at depth y, ρ is fluid density, and h is fluid surface height.
  • Moment about origin for discrete masses
    Moment about origin is \(M_0=\sum m_i x_i\), sum of mass times position.
  • Center of mass for discrete masses on x-axis
    Center of mass \(\bar{x}=\frac{\sum m_i x_i}{\sum m_i}\).
  • Center of mass for continuous mass distribution
    For density δ(x), \(M=\int_a^b \delta(x) dx\) and \(\bar{x}=\frac{\int_a^b x \delta(x) dx}{M}\).
  • Center of mass of a thin plate between curves y=f(x) and y=g(x)
    \(\bar{x}=\frac{1}{M} \int_a^b x \delta [f(x)-g(x)] dx\), \(\bar{y}=\frac{1}{2M} \int_a^b \delta [f^2(x)-g^2(x)] dx\).
  • Moment about y-axis for masses in plane
    Moment about y-axis is \(M_y=\sum m_i x_i\).
  • Moment about x-axis for masses in plane
    Moment about x-axis is \(M_x=\sum m_i y_i\).
  • Center of mass coordinates for masses in plane
    \(\bar{x}=\frac{M_y}{m}, \quad \bar{y}=\frac{M_x}{m}\), where m is total mass.
  • Formula for lateral surface area of a cone
    Lateral surface area = \(\frac{1}{2} \times \text{base circumference} \times \text{slant height}\).
  • Using Mean Value Theorem in arc length derivation
    There exists such that \(\Delta y_i = f'(x_i^*) \Delta x\).
  • Approximating surface area band
    Surface area of band ≈ \(2\pi \frac{y_{i-1}+y_i}{2} \times |P_{i-1}P_i|\).
  • Work formula for pumping liquid
    Work = \(\int \text{weight density} \times \text{volume of slab} \times \text{distance lifted} \, dy\).
  • Fluid force integral setup
    Force = \(\int_a^b w \cdot (\text{depth}) \cdot L(y) \, dy\), where w is weight density.
  • Center of mass of rod with two masses
    Balance condition: \(m_1 d_1 = m_2 d_2\).
  • Definition of moment about an axis
    Moment is sum of mass times distance from the axis, e.g., \(M_y=\sum m_i x_i\).