Final answer:
Finding the definite integral ∫36f(x)dx entails determining the antiderivative of f(x) and then calculating the difference between its values at the bounds 6 and 3, using the Fundamental Theorem of Calculus.
Step-by-step explanation:
To find the definite integral ∫36f(x)dx using the provided information, one must apply the Fundamental Theorem of Calculus. This theorem establishes the relationship between differentiation and integration, specifically that integration can be used to find the sum of infinitesimal quantities over an interval, which can represent areas, volumes, total changes, among others.
The first step is to find the antiderivative, also known as the indefinite integral, of f(x). Then, based on the Fundamental Theorem of Calculus, we evaluate this antiderivative at the upper and lower limits of the integral, in this case, 6 and 3, and subtract the lower limit value from the upper limit value to get the total accumulated change over the interval from 3 to 6.
This process may also apply in contexts such as physics, where the integral calculus can be used to derive kinematic equations for constant acceleration to find velocity and displacement from acceleration. In this case, the areas under the acceleration-time curve represent the velocity change, and the areas under the velocity-time curve represent displacement.