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.

∫ₐ⁰ ƒ(𝓍) d𝓍

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Identify the integral to evaluate: \(\int_{a}^{0} f(x) \, dx\). Notice that the limits of integration go from \(a\) to \(0\), which is in the reverse order of the usual left-to-right direction on the x-axis.

Recall the property of definite integrals that reversing the limits changes the sign: \(\int_{a}^{0} f(x) \, dx = -\int_{0}^{a} f(x) \, dx\).

Look at the graph and observe the area between \(x=0\) and \(x=a\). The shaded region above the x-axis has an area of 16, so \(\int_{0}^{a} f(x) \, dx = 16\) because the function is positive there.

Use the property from step 2 to write \(\int_{a}^{0} f(x) \, dx = -16\) since the integral from 0 to a is positive 16 but the limits are reversed.

Thus, the value of the integral \(\int_{a}^{0} f(x) \, dx\) corresponds to the negative of the area between \(0\) and \(a\) under the curve \(f(x)\).

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주요 개념

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Definite Integral as Net Area

A definite integral over an interval represents the net area between the function's graph and the x-axis. Areas above the x-axis contribute positively, while areas below contribute negatively. This net area interpretation is essential for evaluating integrals from graphs.

Interpreting Areas from Graphs

When given shaded areas on a graph, these represent the absolute values of integrals over subintervals. To find the integral over a larger interval, sum these areas with appropriate signs based on whether the function is above or below the x-axis.

Properties of Definite Integrals and Limits

The integral from a to 0 can be evaluated by reversing limits: ∫ₐ⁰ f(x) dx = -∫₀ᵃ f(x) dx. Understanding how to manipulate integral limits and combine subinterval integrals is crucial for solving problems involving integrals over multiple segments.

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.∫ₐ⁰ ƒ(𝓍) d𝓍