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# Definite Integrals

The definite integral of a function computes the area under the graph of its curve, allowing us to calculate areas and volumes that are not easily done using geometry alone.

# Riemann Sums

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

Evaluate $\int_{-2}^{-1}\left( x^3+3x^2+7 \right)dx + \int_{-1}^{2}\left( y^3+3y^2+7 \right)dy.$

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

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

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

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