Question

The numerical integration of the function $y = 2x + 5$ is carried out between $x = 1$ and $x = 3$, by using ordinates at $x = 1, 2,$ and $3$. Which one of the following statements is TRUE?

• A. Simpson's 1/3 rule will provide exact result but trapezoidal rule will not.
• B. Trapezoidal rule will provide exact result but Simpson's 1/3 rule will not.
• C. Both Simpson's 1/3 and trapezoidal rules will provide exact result.
• D. Neither Simpson's 1/3 rule nor trapezoidal rule will provide exact result.

Hint:

Linear functions always integrate exactly with both trapezoidal and Simpson's 1/3 rule.

Answer 1

The Correct Option is C

The function given is $y = 2x + 5$, which is a first-degree polynomial (a straight line).
The trapezoidal rule gives exact results for all linear functions because the method assumes linear variation between points.
Simpson's 1/3 rule is exact for all polynomials up to third degree, including linear functions.
Thus, Simpson's rule will also give the exact integral for $y = 2x + 5$.
To confirm mathematically:
Exact integral:
\[ \int_{1}^{3} (2x + 5)\, dx = \left[x^2 + 5x\right]_{1}^{3} = (9 + 15) - (1 + 5) = 18 \]
Using trapezoidal rule with $x = 1,2,3$:
\[ \frac{h}{2}\left[y_1 + 2y_2 + y_3\right] = 1\cdot\frac{1}{2}(7 + 2\cdot 9 + 11) = 18 \]
Using Simpson's 1/3 rule:
\[ \frac{h}{3}[y_1 + 4y_2 + y_3] = \frac{1}{3}(7 + 36 + 11) = 18 \]
Both methods give the exact value 18.
Hence, option (C) is the correct statement.