Exercise · ZW-SIMPSON-S-RULE-007 · ★★☆ Intermediate
Using the simple Simpson's rule, what is the approximate value of \(\int_0^1 e^x\,dx\)?
Choose one option.
Show answer
- C. \(\approx 1.719\)
Explanation
With \(h=0.5\), \(f(0)=1\), \(f(0.5)\approx 1.6487\), and \(f(1)\approx 2.7183\), the formula gives \(\frac{1}{6}(1+4\cdot1.6487+2.7183)\approx 1.7189\).
Review the concept: Simpson's rule
Simpson's rule is a numerical quadrature method that approximates \(\int_a^b f(x)\,dx\) by integrating a quadratic polynomial that interpolates \(f\) at \(a\), \(\frac{a+b}{2}\), and \(b\).
\[\int_a^b f(x)\,dx \approx \frac{h}{3}[f(x_0)+4f(x_1)+f(x_2)], h=\frac{b-a}{2}\]
More practice problems: simpson's rule
- ZW-SIMPSON-S-RULE-001 How many subintervals does the composite Simpson's rule require?
- ZW-SIMPSON-S-RULE-002 What polynomial degree does the simple Simpson's rule integrate exactly?
- ZW-SIMPSON-S-RULE-003 Which points are used in the simple Simpson's rule on [a,b]?
- ZW-SIMPSON-S-RULE-004 Which weights appear in the simple Simpson's rule before the factor (h)/(3)?
- ZW-SIMPSON-S-RULE-005 Which derivative appears in the error term of Simpson's rule?
- ZW-SIMPSON-S-RULE-006 Apply the simple Simpson's rule to ∫_0^2 x^2 dx. What is the result?
- ZW-SIMPSON-S-RULE-008 With the composite Simpson's rule and n=4 on [0,4], what is the width h of each subinterval?
- ZW-SIMPSON-S-RULE-009 What is the value of the composite Simpson's rule with n=2 for ∫_0^2 x^3 dx?