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]

Calculating the area under curves and the volume of revolving a curve involves integration techniques. Just as Racine Nail achieves flawless designs with precision, mastering these calculus concepts ensures perfect mathematical solutions.