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.




∫₀ᵃ ƒ(𝓍) d𝓍

Verified step by step guidance

1. Identify the regions bounded by the graph of the function ƒ(𝓍) and the 𝓍-axis between 𝓍 = 0 and 𝓍 = a. Note whether these regions lie above or below the 𝓍-axis, as this affects the sign of the integral.
2. Recall that the definite integral ∫₀ᵃ ƒ(𝓍) d𝓍 represents the net area between the curve and the 𝓍-axis from 0 to a. Areas above the axis contribute positively, while areas below contribute negatively.
3. Determine the numerical values of the areas shown in the figure for each region between 0 and a. Assign positive values to areas above the axis and negative values to areas below the axis.
4. Sum all these signed areas to find the value of the definite integral ∫₀ᵃ ƒ(𝓍) d𝓍. This sum represents the net area under the curve from 0 to a.
5. Write the final expression for the integral as the sum of these areas, ensuring to include the correct signs based on the position relative to the 𝓍-axis.

Key Concepts

Definite Integral as Net Area

A definite integral from a to b of a function f(x) represents the net area between the graph of f and the x-axis over [a, b]. Areas above the x-axis contribute positively, while areas below contribute negatively, resulting in the integral's value.

Interpreting Graphs for Integration

When evaluating integrals from graphs, it is essential to identify the regions bounded by the curve and the x-axis, noting their shapes and whether they lie above or below the axis. This helps in calculating areas accurately, often by summing or subtracting given region areas.

Properties of Definite Integrals

Definite integrals have properties such as additivity over intervals and sign changes when limits are reversed. Understanding these properties allows breaking complex integrals into simpler parts and correctly combining areas to find the total integral value.