Definite integral of polynomial functions

Jelly, \$\frac {30} 7 = 4.29\$

÷J$ŻUḅI

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Takes the polynomial as a list of coefficients in little-endian format (i.e. the example is 7,2,5,4). This can handle any (positive) degree you want, but I've limited it at 30 as the question states \$0 < N < 30\$

+2 bytes and a score of \$3.33\$ if the polynomial must be taken in big-endian format

How it works

÷J$ŻUḅI - Main link. Takes coeffs on the left and [a, b] on the right
  $     - To the coeffs:
 J      -   Yield [1, 2, 3, ..., N]
÷       -   Divide each coeff by the orders
   Ż    - Prepend 0
    U   - Reverse
     ḅ  - Calculate as polynomials, with x = a and x = b
      I - Reduce by subtraction

J, \$\frac{30}{12}\approx2.5\$

[:-/[-p..p.[

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-1 thanks to Bubbler

• p.. Integral of a polynomial. Returns a list representing the solution polynomial.
• p. Evaluate polynomial at given bounds.
• [- Subtract from constant terms (makes both values negative).
• -/ And subtract negative ending bound answer from negative starting bound answer.

Factor, \$\frac{30}{56}\approx0.5357\$

[ [ 1 + 3dup nip v^n first2 - swap / * ] map-index sum ]

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Takes the bounds and coefficients in reverse, e.g. { 3 2 } { 7 2 5 4 }, and returns a rational number, e.g. 108+2/3.

To take the inputs as given and return a float, add a reverse, a neg, and a >float to get 75 bytes:

Factor, \$\frac{30}{75}=0.4\$

[ reverse [ 1 + 3dup nip v^n first2 - swap / * ] map-index sum neg >float ]

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