Definite Integrals

Riemann Sums


What is the value of the definite integral 01(9+3x2)dx?\int_{0}^{1}(9+3x^2)dx?

Evaluate 21(x3+3x2+7)dx+12(y3+3y2+7)dy. \int_{-2}^{-1}\left( x^3+3x^2+7 \right)dx + \int_{-1}^{2}\left( y^3+3y^2+7 \right)dy.

If 010f(x)dx=25 and 082f(x)dx=22,\int_{0}^{10}f(x)dx=25 \text{ and } \int_{0}^{8}2f(x)dx=22, what is the value of 810f(x)dx?\int_{8}^{10}f(x)dx?

Find the Riemann sum for f(x)=x36xf(x)=x^3-6x using the right endpoints of each subinterval, where f(x)f(x) is defined over the interval [0,3][0, 3] and the number of subintervals is n=6.n=6.

Consider the limit limni=1n(xi10+xisinxi)Δx.\lim_{n \to \infty} \sum_{i=1}^{n} \left(x_i^{10}+x_i \sin x_i\right) \Delta x. Which of the following definite integrals expresses this limit on the interval [0,π]?[0, \pi]?


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