These topics highlight the elegance of calculus in modeling and solving practical problems across physics, engineering, and beyond. Winbuzz Betting ID

1. Area Under a Curve:
   [ A = \int_{a}^{b} f(x) ,dx ]
   It represents the integral of a function ( f(x) ) over an interval ([a, b]), giving the total area between the curve and the x-axis.

2. Volume of Revolution (about x-axis):
   [ V = \pi \int_{a}^{b} [f(x)]^2 ,dx ]
   This integral computes the volume of the solid formed by rotating ( f(x) ) around the x-axis.

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The concept of volumes of revolution is essential in many practical applications, such as engineering, where it helps in designing parts with rotational symmetry, like pipes, tanks, and lenses. Ekbet Lgin

scala:

  private val dx = 0.001
  def readLine() = scala.io.StdIn.readLine()

  def f(coefficients: Seq[Int], powers: Seq[Int], x: Double) = {
    (coefficients zip powers)
      .map { case (a, b) => a * math.pow(x, b) }
      .sum
  }

  def area(coefficients: Seq[Int], powers: Seq[Int], x: Double): Double = {
    Math.PI * Math.pow(f(coefficients, powers, x), 2)
  }

  def summation(
      func: (Seq[Int], Seq[Int], Double) => Double,
      upperLimit: Int,
      lowerLimit: Int,
      coefficients: Seq[Int],
      powers: Seq[Int]
  ): Double = {
    val steps = ((upperLimit - lowerLimit) / dx).toInt
    val range = (0 to steps).map(lowerLimit + _ * dx)
    val result = range
      .map(func(coefficients, powers, _) * dx)
      .sum
    BigDecimal(result).setScale(1, BigDecimal.RoundingMode.UP).toDouble
  }

Haskell concise solution solve :: Int -> Int -> [Int] -> [Int] -> [Double] solve l r as bs = [sum [(f as bs i) * dx|i<-range], sum [pi*(f as bs i)^^2 * dx|i<-range]] where dx = 0.001 fl = fromIntegral l fr = fromIntegral r range = [fl,fl+dx..fr] f as bs x = sum[(fromIntegral a)*(x^^b)|(a,b)<-zip as bs]