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Calculus

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

Riemann Sums and Definite Integrals

Find the approximate value of $\displaystyle{\int_0^{2} 7x^{2}dx}$ using a right Riemann sum by dividing the interval into $4$ pieces.

\frac{1683}{64}

\frac{1681}{64}

\frac{841}{32}

\frac{105}{4}

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by Brilliant Staff

Which of the following represents the approximation of $\displaystyle{\int_{0}^{4}x^{5}dx}$ using a left Riemann sum?

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

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

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

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

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by Brilliant Staff

Find the approximate value of $\displaystyle{\int_0^1 (6x^2+2)dx}$ using a right Riemann sum, by dividing the interval into $7$ parts.

\frac{218}{49}

\frac{216}{49}

\frac{31}{7}

\frac{215}{49}

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by Brilliant Staff

The following is Alex's approximation of an integration by using a right Riemann sum: $\frac{9}{5}\cdot\left(\left(\frac{3}{5}\right)^{7}+\left(\frac{6}{5}\right)^{7}+\left(\frac{9}{5}\right)^{7}+\left(\frac{12}{5}\right)^{7}+\left(\frac{15}{5}\right)^{7}\right).$ Which of the following integrals is Alex approximating?

\displaystyle{\int_0^{7}{3}x^{5}dx}

\displaystyle{\int_0^{7}{3}x^{4}dx}

\displaystyle{\int_0^{3}{3}x^{7}dx}

\displaystyle{\int_0^{3}{3}x^{8}dx}

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by Brilliant Staff

What is the Riemann sum of the function $f(x)= x^3-6x$ is in the interval $[0, 6]$ , if we divide it into 3 equal parts and use the midpoint of each interval?

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by Brilliant Staff