With respect to the numerical evaluation of the definite integral. $K={\int }_a^b{x}^{2}dx,$ where a and b are given, which of the following statements is/are TRUE?

I) The value of K obtained using the trapezoidal rule is always greater than or equal to the exact value of the definite integral.

II) The value of K obtained using the Simpson’s rule is always equal to the exact value of the definite integral.

This question asks us to evaluate the truthfulness of two statements concerning the numerical approximation of the definite integral $K={\int }_a^b{x}^{2}dx$. We need to check the accuracy of the Trapezoidal rule and Simpson's rule for the function $f(x)=x^2$.

**Definite Integral $K={\int }_a^b{x}^{2}dx$ Analysis**

The definite integral $K={\int }_a^b{x}^{2}dx$ represents the area under the curve $y=x^2$ between the limits $x=a$ and $x=b$. We will analyze how the Trapezoidal rule and Simpson's rule approximate this value.

**Trapezoidal Rule and $f(x)=x^2$ Concavity

Statement I:** The value of K obtained using the trapezoidal rule is always greater than or equal to the exact value of the definite integral.

The Trapezoidal rule approximates the area under a curve by dividing it into several trapezoids. The accuracy of the Trapezoidal rule depends on the concavity of the function.

For the function $f(x)=x^2$, let's find its second derivative:

- First derivative: $f^{\prime }(x)=\frac{d}{dx}(x^2)=2x$
- Second derivative: $f^{\prime \prime }(x)=\frac{d}{dx}(2x)=2$

Since the second derivative, $f^{\prime \prime }(x)=2$, is positive for all values of $x$, the function $f(x)=x^2$ is concave up everywhere. When a function is concave up, the straight line segment connecting two points on the curve lies above the curve itself. Consequently, the trapezoids formed by approximating the area with straight line segments will enclose a larger area than the actual area under the curve.

Therefore, the Trapezoidal rule will always overestimate the definite integral for a concave up function. This means the value obtained using the Trapezoidal rule is greater than the exact value.

Thus, Statement I is TRUE.

Simpson's Rule Exactness for Polynomials

Statement II: The value of K obtained using the Simpson’s rule is always equal to the exact value of the definite integral.

Simpson's rule is another numerical method for approximating definite integrals. It uses parabolic segments instead of straight lines to approximate the curve. A key property of Simpson's rule is its accuracy for polynomial functions.

Simpson's rule is known to be exact for approximating the integral of any polynomial function of degree up to 3 (i.e., cubic polynomials).

In this case, the function is $f(x)=x^2$, which is a polynomial of degree 2.

Since the degree of the polynomial ($2$) is less than or equal to $3$, Simpson's rule will provide the exact value of the definite integral $K={\int }_a^b{x}^{2}dx$.

Thus, Statement II is TRUE.

Statements TRUE Status

Based on the analysis of the function $f(x)=x^2$ and the properties of the Trapezoidal rule and Simpson's rule:

- Statement I is TRUE because $f(x)=x^2$ is concave up ($f^{\prime \prime }(x)>0$), causing the Trapezoidal rule to overestimate.
- Statement II is TRUE because Simpson's rule is exact for polynomials of degree up to 3, and $f(x)=x^2$ is a degree 2 polynomial.

Therefore, both statements are true.