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P(0,3), Q(0.5,4) and R(1,5) are three points on the curve defined by f(x). Numerical integration is carried out using both trapezoidal rule and simpson’s rule within limits x = 0 and x = 1 for the curve. The difference between the two results will be
  • a)
    0
  • b)
    0.25
  • c)
    0.5
  • d)
    1
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
P(0,3), Q(0.5,4) and R(1,5) are three points on the curve defined by f...
Explanation:

Trapezoidal Rule:
- Trapezoidal rule is based on approximating the curve by trapezoids and summing the areas of these trapezoids to estimate the integral.
- The formula for trapezoidal rule is:
\[ \int_{a}^{b} f(x)dx \approx \frac{h}{2}[f(a) + 2f(a+h) + 2f(a+2h) + ... + f(b)] \]
- Using trapezoidal rule with the given points P(0,3), Q(0.5,4), and R(1,5) will result in a certain numerical value.

Simpson's Rule:
- Simpson's rule is based on approximating the curve by quadratic polynomials and summing the areas of these polynomials to estimate the integral.
- The formula for Simpson's rule is:
\[ \int_{a}^{b} f(x)dx \approx \frac{h}{3}[f(a) + 4f(a+h) + 2f(a+2h) + 4f(a+3h) + ... + f(b)] \]
- Using Simpson's rule with the given points P(0,3), Q(0.5,4), and R(1,5) will result in a certain numerical value.

Difference between the two results:
- The difference between the results obtained using trapezoidal rule and Simpson's rule will be zero if the curve is a quadratic polynomial.
- Since the points P(0,3), Q(0.5,4), and R(1,5) lie on a curve defined by f(x), and both trapezoidal rule and Simpson's rule are used within the same limits, the difference between the two results will be zero.
Therefore, the correct answer is option 'A'.
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P(0,3), Q(0.5,4) and R(1,5) are three points on the curve defined by f(x). Numerical integration is carried out using both trapezoidal rule and simpson’s rule within limits x = 0 and x = 1 for the curve. The difference between the two results will bea)0b)0.25c)0.5d)1Correct answer is option 'A'. Can you explain this answer?
Question Description
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