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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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Related Practice

Textbook Question

Average values Find the average value of the following functions on the given interval. Draw a graph of the function and indicate the average value.

ƒ(𝓍) = 𝓍ⁿ on [0,1] , for any positive integer n

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Textbook Question

Average values Find the average value of the following functions on the given interval. Draw a graph of the function and indicate the average value.

ƒ(𝓍) = 𝓍³ on [―1, 1]

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