Calculate Integral Polynomial by Slicing

What is the Slicing Method?

The slicing method, also known as the Riemann sum approximation, is a way to estimate the area under a curve by dividing it into vertical slices. Each slice represents a small rectangle whose height is determined by the function's value at that point, and whose width is the small change in x (Δx).

As the number of slices increases and their width decreases, the approximation becomes more accurate. In the limit, as Δx approaches zero, the sum of the areas of the slices becomes the exact integral of the function.

How to Calculate Polynomial Integrals

To calculate the integral of a polynomial using the slicing method:

1. Define the polynomial function f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀.
2. Choose a range [a, b] over which to integrate.
3. Divide the interval [a, b] into n equal subintervals, each of width Δx = (b - a)/n.
4. Evaluate the function at the left endpoint of each subinterval: f(x₀), f(x₁), ..., f(xₙ₋₁).
5. Multiply each function value by Δx to get the area of each slice: Δx * f(x₀), Δx * f(x₁), ..., Δx * f(xₙ₋₁).
6. Sum all the slice areas to approximate the integral: Σ (Δx * f(xᵢ)) from i=0 to n-1.
7. As n increases, the approximation becomes more accurate.

For a polynomial, the exact integral can also be found using the power rule for integration, but the slicing method provides a practical way to understand the concept.

The Formula

The integral of a polynomial f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀ from a to b using the slicing method is approximated by:

The exact integral of the polynomial can be calculated using the power rule:

Worked Example

Let's calculate the integral of f(x) = 3x² + 2x + 1 from 0 to 2 using the slicing method with n=4 slices.

1. Δx = (2 - 0)/4 = 0.5
2. Evaluate f(x) at x₀=0, x₁=0.5, x₂=1.0, x₃=1.5:
   • f(0) = 3(0)² + 2(0) + 1 = 1
   • f(0.5) = 3(0.25) + 2(0.5) + 1 = 1.75 + 1 + 1 = 3.75
   • f(1.0) = 3(1) + 2(1) + 1 = 3 + 2 + 1 = 6
   • f(1.5) = 3(2.25) + 2(1.5) + 1 = 6.75 + 3 + 1 = 10.75
3. Calculate slice areas:
   • 0.5 * 1 = 0.5
   • 0.5 * 3.75 = 1.875
   • 0.5 * 6 = 3
   • 0.5 * 10.75 = 5.375
4. Sum the areas: 0.5 + 1.875 + 3 + 5.375 = 10.75

The approximation of the integral is 10.75. The exact integral using the power rule is:

As expected, the approximation improves as we increase the number of slices.