Calculus
Definite Integrals
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 \([0, \pi]?\)