When using a change of variables \( u=g(x) \) to evaluate the definite integral \( \int_{a}^{b} \) fig(x) \( g g^{\prime}(x) \), \( x \), how are the limits of intiegration fransformed? Choose the correct answer beiove A. The lower limit of integration becomes sta) and the upper limit of infegration beccomes g(b). B. The lower limit of integration becomes f(a) and the upper limit of infegration becomes fis) C. The lower limit of integration becomes \( g^{\prime}(a) \) and the upper limit of integration beccmes \( g^{\prime \prime}(\mathrm{C}) \) ) D. The upper and lower limits do not change.

When you perform a change of variables \( u = g(x) \) to evaluate a definite integral, the limits of integration are transformed based on the transformation function. Specifically, the lower limit of integration changes to \( g(a) \), and the upper limit of integration changes to \( g(b) \). So the correct answer aligns with option A: The lower limit of integration becomes \( g(a) \) and the upper limit of integration becomes \( g(b) \). Now, let's delve deeper into this topic! Changing variables in integration allows us to simplify complex integrals by transforming them into more manageable forms, especially when the new variable corresponds to a simpler function. This technique is widely used in calculus to tackle integrals that initially seem daunting, making it an essential tool for mathematicians and scientists alike. Moreover, it’s crucial to remember to also adjust the differential accordingly during substitution. The integral transforms not only the variable of integration but also the differential, meaning the entire expression inside the integral changes, reflecting the characteristics of the new variable effectively. This ensures accurate evaluation and helps avoid common pitfalls!