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.

"Exploring the intricacies of calculus through understanding the area under curves and the volume obtained by revolving them offers profound insights into mathematical abstraction and real-world applications." Cricbet99 Login ID and Password