Combining Integrals with Equal and Contiguous Ranges

Introduction to Combining Integrals

In definite integration, there are several ways to merge multiple integrals into a single, more manageable expression. This is particularly useful in the ESAT when evaluating the total area under a curve or simplifying complex algebraic strings of integrals. There are two primary situations where this is possible: when the ranges (the limits of integration) are identical, and when the ranges are contiguous.

Combining Integrals with Equal Ranges

When two or more integrals share the exact same lower and upper limits, they can be combined into a single integral of the sum or difference of the functions. This property arises because integration is a linear operation. You can integrate functions term by term or combine them first and then integrate the entire expression. The general rule is:

$\int_{a}^{b} f(x)dx + \int_{a}^{b} g(x)dx = \int_{a}^{b} [f(x) + g(x)]dx$

For example, if you are given $\int_{2}^{5} f(x)dx + \int_{2}^{5} g(x)dx$, you can simplify this to $\int_{2}^{5}[f(x) + g(x)]dx$. This is often much faster than evaluating two separate integrals, especially if the combined function $f(x) + g(x)$ simplifies algebraically before you begin the integration process.

Combining Integrals with Contiguous Ranges

Integrals can also be combined if the upper limit of one integral is the lower limit of the next. This allows you to join the ranges together into one continuous interval. The general rule is:

$\int_{a}^{b} f(x)dx + \int_{b}^{c} f(x)dx = \int_{a}^{c} f(x)dx$

Consider the following example: $\int_{2}^{4} f(x)dx + \int_{4}^{5} f(x)dx = \int_{2}^{5} f(x)dx$. Here, the first integral covers the interval from 2 to 4, and the second continues from 4 to 5. When added, they cover the entire interval from 2 to 5. This property is a direct application of the fact that the total area under a curve over a long interval is equal to the sum of the areas over its sub-intervals.

Using the Fundamental Theorem of Calculus to Simplify Expressions

The Fundamental Theorem of Calculus (FTC) provides the formal justification for these combinations. It states that if $F(x)$ is the antiderivative of $f(x)$, then $\int_{a}^{b} f(x)dx = F(b) - F(a)$. We can use this to unpack and verify complex expressions where limits might not be in numerical order. It is worth checking that you have both an intuitive grasp of how these areas combine and a formal understanding of the algebra involved.

Consider this specific case from the guide: $\int_{2}^{4} f(x) dx + \int_{4}^{3} f(x) dx = \int_{2}^{3} f(x) dx$. At first, this looks unusual because the second integral goes backwards from 4 to 3. However, we can use the FTC to see why this works by inserting an intermediate point, such as 3, into the first integral to show the following steps:

1. Split the first integral at the point 3: $\int_{2}^{4} f(x) dx = \int_{2}^{3} f(x) dx + \int_{3}^{4} f(x) dx$.
2. Recognise that $\int_{4}^{3} f(x) dx$ is the negative of $\int_{3}^{4} f(x) dx$.
3. Combine them: $\int_{2}^{4} f(x) dx + \int_{4}^{3} f(x) dx = \int_{2}^{3} f(x) dx + \int_{3}^{4} f(x) dx - \int_{3}^{4} f(x) dx = \int_{2}^{3} f(x) dx$.

The middle terms cancel out perfectly, leaving only the integral from 2 to 3. Mastering this limit manipulation is a vital skill for solving advanced calculus problems efficiently.