Definite integrals from graphs The figure shows the areas of regions bounded by the graph of ƒ and the 𝓍-axis. Evaluate the following integrals.




∫₀ᶜ |ƒ(𝓍)| d𝓍

Verified step by step guidance

1. Identify the interval from 0 to c on the x-axis and observe the graph of the function ƒ(𝓍) over this interval.
2. Since the integral is of the absolute value |ƒ(𝓍)|, note that all areas between the graph and the x-axis will be considered positive, regardless of whether ƒ(𝓍) is above or below the x-axis.
3. Break the interval [0, c] into subintervals where the function ƒ(𝓍) does not change sign (i.e., where it is either entirely positive or entirely negative).
4. For each subinterval, calculate the area between the graph of ƒ(𝓍) and the x-axis. If the function is negative on that subinterval, take the absolute value of the integral (which corresponds to the area).
5. Sum all these positive areas from each subinterval to find the value of the definite integral ∫₀ᶜ |ƒ(𝓍)| d𝓍.

Key Concepts

Definite Integral

A definite integral calculates the net area under a curve between two points on the x-axis. It sums the values of the function over the interval, considering areas above the x-axis as positive and below as negative. This concept is fundamental for interpreting integrals from graphs.

Absolute Value of a Function in Integration

Integrating the absolute value of a function means summing all areas as positive, regardless of whether the function is above or below the x-axis. This ensures the integral represents total area without cancellation from negative parts, which is crucial when evaluating ∫₀ᶜ |ƒ(x)| dx.

Area Interpretation from Graphs

When given a graph, the definite integral corresponds to the area between the curve and the x-axis. Understanding how to read and calculate these areas, especially when the function crosses the axis, helps in evaluating integrals accurately, including those involving absolute values.