Definite Integrals Demystified: Tables, Properties, and an Oil Consumption Case

Basic Integral Table

A basic integral table is provided as a quick reference for common antiderivatives.

Properties of Definite Integrals

Addition and Subtraction

Definite integrals are linear with respect to addition and subtraction: ∫[a,b](f(x)±g(x))dx = ∫[a,b]f(x)dx ± ∫[a,b]g(x)dx .

Scalar Multiplication

Multiplying a function by a constant scales the integral: ∫[a,b]c·f(x)dx = c·∫[a,b]f(x)dx .

Case Study

In recent decades, global oil consumption has grown exponentially at an approximate rate of 0.07 per year. In early 1970, consumption was about 16.1 billion barrels. Let R(t) denote the annual oil consumption rate t years after 1970. Using the given exponential model, calculate the total oil consumption from 1970 to 1990.

Let T(t) represent the cumulative oil consumption from 1970 (t = 0) to year t. Since the consumption rate is R(t), we have T′(t)=R(t). Integrating the rate over the interval yields the total consumption.

Applying the method of differential approximation, we set up the integral and evaluate it to obtain the total oil used between 1970 and 1990.