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.




∫ₐᶜ ƒ(𝓍) d𝓍

Verified step by step guidance

1. Identify the points a and c on the x-axis from the graph, which define the limits of integration for the integral \( \int_{a}^{c} f(x) \, dx \).
2. Observe the graph of \( f(x) \) between \( x = a \) and \( x = c \) and note the areas bounded by the curve and the x-axis. Pay attention to whether the graph is above or below the x-axis in each region, as this affects the sign of the integral.
3. Calculate the area of each region between the curve and the x-axis separately. If the graph is above the x-axis, the area contributes positively to the integral; if below, it contributes negatively.
4. Sum the signed areas of all regions between \( a \) and \( c \) to find the value of the definite integral \( \int_{a}^{c} f(x) \, dx \).
5. Express the final integral as the algebraic sum of these areas, which represents the net area between the curve and the x-axis over the interval \( [a, c] \).

Key Concepts

Definite Integral as Net Area

A definite integral represents the net area between the graph of a function and the x-axis over a given interval. Areas above the x-axis contribute positively, while areas below contribute negatively, affecting the integral's value.

Interpreting Graphical Areas

When evaluating integrals from graphs, it is essential to identify and measure the areas of regions bounded by the curve and the x-axis. Understanding which parts lie above or below the axis helps determine the sign and magnitude of each area.

Properties of Definite Integrals

Definite integrals are additive over adjacent intervals, meaning the integral from a to c can be split into integrals over subintervals. This property allows summing individual areas to find the total integral value.