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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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