The rate of consumption of oil (in billions of barrels) by...

The rate of consumption of oil (in billions of barrels) by the company in Exercise 71 was given as
$$1.2 e^{0.04 t}$$
where $t=0$ corresponds to 2010 . Find the total amount of oil used by the company from 2010 to year $T$ . At this rate, how much will be used in 5 years?

Key Concepts

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links the concepts of differentiation and integration. It states that if a function is continuous on an interval, then its definite integral over that interval can be computed using any of its antiderivatives. This theorem essentially provides the method to calculate the total accumulation from a given rate of change.

Antiderivative

The antiderivative of a function is a function whose derivative is the original function. In the context of definite integrals, finding the antiderivative is a crucial step because it allows for the application of the Fundamental Theorem of Calculus to compute the accumulated quantity between two limits.

Exponential Functions

Exponential functions model processes that change at rates proportional to their current values. In calculus, these functions are fundamental for describing growth or decay phenomena and are characterized by a constant base raised to a variable power. Their properties, such as constant relative rate of change, make them useful in various contexts including population growth, radioactive decay, and financial interest calculations.

Definite Integration

Definite integration is the process of calculating the accumulation of quantities, such as areas under curves or total amounts over time, by summing an infinite number of infinitesimally small contributions. It provides a precise method to convert a rate function into a total quantity, typically using specified limits of integration to denote the interval of interest.

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