3. \( \operatorname{Let} x=-\frac{5}{6} \) and \( y=\frac{4}{3} \quad \int_{-\frac{8}{8}}^{x} \)

Let’s compute the definite integral from \(-1\) to \(-\frac{5}{6}\). For simplicity, assume we're integrating a function like \(f(t) = t\). The integral \( \int_{-1}^{-\frac{5}{6}} t \, dt \) gives us: \[ = \left[ \frac{t^2}{2} \right]_{-1}^{-\frac{5}{6}} = \frac{(-\frac{5}{6})^2}{2} - \frac{(-1)^2}{2} = \frac{\frac{25}{36}}{2} - \frac{1}{2} = \frac{25}{72} - \frac{36}{72} = -\frac{11}{72}. \] Now, wouldn’t you love to know how integrals have been a cornerstone in mathematics since the days of Archimedes, who pioneered the use of infinitesimal areas? Isn’t it remarkable that concepts we've built upon for centuries still drive modern calculus today? You might also find yourself using definite integrals not just in academia but in real-world scenarios like calculating areas under curves or even for determining quantities like distance or mass. Just think about how engineers and scientists use these tools to predict everything from motion in physics to probabilities in statistics!