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?
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?
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Linear functions always integrate exactly with both trapezoidal and Simpson's 1/3 rule.
Updated On: Jan 13, 2026
- Simpson's 1/3 rule will provide exact result but trapezoidal rule will not.
- Trapezoidal rule will provide exact result but Simpson's 1/3 rule will not.
- Both Simpson's 1/3 and trapezoidal rules will provide exact result.
- Neither Simpson's 1/3 rule nor trapezoidal rule will provide exact result.
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The Correct Option is C
Solution and Explanation
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.
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.
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