Numerical Approximation of Integrals

Concept Quizzes

Riemann Sums and Definite Integrals
Integral Approximation - Trapezium Rule
Integral Approximation - Simpson's Rule
Integral Approximation - Accuracy of Approaches
Integral Approximation - Increasing Intervals

Challenge Quizzes

Numerical Approximation of Integrals: Level 4 Challenges

Integral Approximation - Trapezium Rule

The following is Devin's approximation of an integral using the Trapezium rule: $\frac{6}{5}\cdot\left(0 + 2\cdot\left(\frac{3}{5}\right)^{5}+2\cdot\left(\frac{6}{5}\right)^{5}+2\cdot\left(\frac{9}{5}\right)^{5}+2\cdot\left(\frac{12}{5}\right)^{5}+\left(\frac{15}{5}\right)^{5}\right).$ Which of the following integrals is Devin approximating?

\displaystyle{\int_0^{5}{4}x^{5}dx}

\displaystyle{\int_0^{3}{4}x^{5}dx}

\displaystyle{\int_0^{5}{4}x^{4}dx}

\displaystyle{\int_0^{3}{4}x^{6}dx}

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Find the approximate value of $\displaystyle{\int_0^{2} 7x^{3}dx}$ using the Trapezium rule, by dividing the interval into $4$ pieces.

\frac{7617}{256}

\frac{119}{4}

\frac{3809}{128}

\frac{7615}{256}

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What is the Trapezium rule approximation to the definite integral $\displaystyle{\int_{-1}^1 (10-2x^2)dx}$ using four intervals?

\frac{37}{2}

\frac{78}{8}

\frac{70}{8}

\frac{41}{2}

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Which of the following represents the approximation of $\displaystyle{\int_{0}^{2}x^{4}dx}$ using the Trapezium rule?

\displaystyle{\frac{16}{n^{5}}\sum_{k=1}^{n}k^{4}}

\displaystyle{\frac{16}{n^{5}}\sum_{k=1}^{n}\left((k-1)^{4}+k^{4}\right)}

\displaystyle{\frac{16}{n^{5}}\sum_{k=0}^{n-1}\left((k-1)^{4}+k^{4}\right)}

\displaystyle{\frac{16}{n^{5}}\sum_{k=0}^{n-1}k^{4}}

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Find the approximate value of $\displaystyle{\int_0^1 (12x^2+2)dx}$ using the Trapezium rule, by dividing the interval into $5$ parts.

\frac{91}{15}

\frac{457}{75}

\frac{152}{25}

\frac{454}{75}

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by Brilliant Staff