Consider the definite integral

${\int }_1^2(4x^2+2x+6)dx$

Let I_e be the exact value of the integral. If the same integral is estimated using Simpson’s rule with 10 equal subintervals, the value is I_s. The percentage error is defined as e = 100 × (I_e - I_s)/I_e The value of e is

The correct answer is 0.

To understand why the percentage error is exactly 0, let us look at the mathematical properties of Simpson's rule.

Simpson's rule is a method for numerical integration that approximates the integrand using quadratic polynomials (paraboloids) over each pair of subintervals. Specifically, for a function $f\left(x\right)$, the error term $E$ associated with Simpson's rule over an interval $[a,b]$ using $n$ subintervals is given by:

$E=-\frac{(b-a{)}^5}{180n^4}{f}^{\left(4\right)}\left(\xi \right)$

where $\xi$ is some number in the interval $[a,b]$, and $f^{\left(4\right)}\left(x\right)$ is the fourth derivative of the function $f\left(x\right)$ with respect to $x$.

An important consequence of this error formula is that if the fourth derivative of the function is zero everywhere on the interval, the error of the approximation is exactly zero. This means that Simpson's rule is exact (gives zero error) for any polynomial of degree 3 or less.

Let us analyze the integrand in our given definite integral:
$f\left(x\right)=4x^2+2x+6$

This is a quadratic polynomial (degree 2). Let us compute its successive derivatives:

First derivative:
$f^{\text{'}}\left(x\right)=8x+2$

Second derivative:
$f^{\text{'}\text{'}}\left(x\right)=8$

Third derivative:
$f^{\text{'}\text{'}\text{'}}\left(x\right)=0$

Fourth derivative:
$f^{\left(4\right)}\left(x\right)=0$

Since the fourth derivative $f^{\left(4\right)}\left(x\right)$ is identically zero, the error term $E$ is also zero. Therefore, the estimated value using Simpson's rule ($I_s$) is exactly equal to the exact value of the integral ($I_e$).

Since $I_e=I_s$, the percentage error is:
$e=100\times \frac{I_e-I_s}{I_e}=100\times \frac{0}{I_e}=0$

Thus, the value of the percentage error $e$ is 0.