Definite integral of polynomial functions
Jelly, \$\frac {30} 7 = 4.29\$
÷J$ŻUḅI

Try it online!
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 &lt; N &lt; 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