Exercise · ZW-SIMPSON-S-RULE-009 · ★★☆ Intermediate
What is the value of the composite Simpson's rule with \(n=2\) for \(\int_0^2 x^3\,dx\)?
Choose one option.
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- A. \(4\)
Explanation
With \(h=1\), \(f(0)=0\), \(f(1)=1\), \(f(2)=8\): \(\frac{1}{3}(0+4+8)=4\), exact for cubics.
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-007 Using the simple Simpson's rule, what is the approximate value of ∫_0^1 e^x dx?
- 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?