Definite Integrals with Modulus: Splitting at Zero Crossings

Concept Overview

This question tests the ability to evaluate definite integrals of functions containing absolute value expressions. The key technique involves identifying the points where the expression inside the modulus becomes zero, as these are the points where the function's definition changes. By splitting the integral at these "zero crossing" points, we can rewrite the integral as a sum of simpler integrals, each with a well-defined integrand, allowing for straightforward evaluation.

Worked Solution

Step 1: Identify the points where the expression inside the modulus is zero. For the expression $|f(x)|$, we need to find the values of $x$ such that $f(x) = 0$. These points will be our potential splitting points for the integral.

Step 2: Determine the sign of $f(x)$ in the intervals defined by the zero crossing points and the limits of integration. For each interval $(a, b)$ created by the zero crossing points and the original limits of integration, we need to determine if $f(x) > 0$ or $f(x) < 0$. This will tell us whether $|f(x)| = f(x)$ or $|f(x)| = -f(x)$ in that interval.

Step 3: Split the definite integral at the identified zero crossing points that lie within the integration limits. If the integration is from $L$ to $U$, and a zero crossing point $c$ is such that $L < c < U$, then the integral $\int_L^U |f(x)| dx$ can be split as $\int_L^c |f(x)| dx + \int_c^U |f(x)| dx$.

Step 4: Rewrite the integral with the appropriate sign for the modulus function in each sub-interval. Using the information from Step 2, replace $|f(x)|$ with $f(x)$ or $-f(x)$ in each of the split integrals. For example, if $f(x) \ge 0$ for $x \in [L, c]$ and $f(x) < 0$ for $x \in [c, U]$, then the integral becomes $\int_L^c f(x) dx + \int_c^U (-f(x)) dx$.

Step 5: Evaluate each of the resulting definite integrals. Now that the absolute value is removed, each integral can be evaluated using standard integration techniques. Find the antiderivative of the integrand and apply the Fundamental Theorem of Calculus.

Step 6: Sum the results of the evaluated integrals. The final answer is the sum of the values obtained in Step 5.

Example: Evaluate $\int_{-1}^2 |x^2 - 1| dx$.

Step 1: Find the zero crossings for $x^2 - 1$. $x^2 - 1 = 0 \implies (x-1)(x+1) = 0 \implies x = 1$ or $x = -1$.

Step 2: Determine the sign of $x^2 - 1$ in the relevant intervals. The limits of integration are $-1$ and $2$. The zero crossing points are $-1$ and $1$. The intervals to consider are $[-1, 1]$ and $[1, 2]$.

• For $x \in (-1, 1)$, let's test $x=0$: $0^2 - 1 = -1 < 0$. So, $x^2 - 1 < 0$ in this interval.
• For $x \in (1, 2)$, let's test $x=1.5$: $(1.5)^2 - 1 = 2.25 - 1 = 1.25 > 0$. So, $x^2 - 1 > 0$ in this interval.

Step 3: Split the integral. The zero crossing point $x=1$ lies within the integration limits $[-1, 2]$. The point $x=-1$ is one of the limits. So we split at $x=1$:

Step 4: Rewrite with appropriate signs.

• In $[-1, 1]$, $x^2 - 1 \le 0$, so $|x^2 - 1| = -(x^2 - 1) = 1 - x^2$.
• In $[1, 2]$, $x^2 - 1 \ge 0$, so $|x^2 - 1| = x^2 - 1$. The integral becomes:

Step 5: Evaluate each integral. First integral:

Second integral:

Step 6: Sum the results.

Key Takeaways:

• The absolute value function $|f(x)|$ is equal to $f(x)$ when $f(x) \ge 0$ and $-f(x)$ when $f(x) < 0$.
• Zero crossing points of $f(x)$ are critical for splitting the integration interval.
• Ensure that the splitting points lie strictly between the original limits of integration.
• Carefully determine the sign of $f(x)$ in each sub-interval to correctly remove the modulus.

Answer: $\boxed{\frac{8}{3}}$