In Simpson's one-third rule the integrand `int_a^b f(x) dx `assumes the shape of a curve given by

Options

A. Parabola
B. hyperbola
C. circle
D. none of these

Answer

To solve the question regarding the shape of the curve assumed by the integrand in Simpson's one-third rule, we can follow these steps:

### Step 1: Understand Simpson's One-Third Rule
Simpson's one-third rule is a numerical method used to approximate the definite integral of a function. It is based on the idea of approximating the integrand (the function being integrated) by a polynomial, specifically a quadratic polynomial (which is a parabola).

### Step 2: Identify the Polynomial Approximation
In Simpson's one-third rule, we approximate the function \( f(x) \) over the interval \([a, b]\) using a quadratic polynomial. This polynomial is determined by the values of the function at three points: \( f(a) \), \( f\left(\frac{a+b}{2}\right) \), and \( f(b) \).

### Step 3: Recognize the Shape of the Curve
Since the approximation involves a quadratic polynomial, the shape of the curve that represents the integrand \( f(x) \) in this method is a parabola.

### Step 4: Conclusion
Based on the understanding that Simpson's one-third rule uses a quadratic polynomial to approximate the integrand, we conclude that the correct answer is:

**The integrand \( \int_a^b f(x) dx \) assumes the shape of a parabola.**

### Final Answer
**Parabola**

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