Riemann Sums and Definite Integrals Practice Problems Online

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

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

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

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.

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?

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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