Textbook Question
Area versus net area Find (i) the net area and (ii) the area of the region bounded by the graph of ƒ and the 𝓍-axis on the given interval. You may find it useful to sketch the region.
ƒ(𝓍) = 𝓍⁴ ― 𝓍² on [―1, 1]
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9. Graphical Applications of Integrals
Area Between Curves
2:17 minutesProblem 5.3.14d
Textbook Question
Area functions The graph of ƒ is shown in the figure. Let A(x) = ∫₀ˣ ƒ(t) dt and F(x) = ∫₂ˣ ƒ(t) dt be two area functions for ƒ. Evaluate the following area functions.
(d) F(8)
Verified step by step guidance1
Step 1: Understand the problem. We are tasked with evaluating F(8), where F(x) = ∫₂ˣ ƒ(t) dt. This represents the net area under the curve of ƒ(t) from t = 2 to t = 8.
Step 2: Analyze the graph. The graph shows the function ƒ(t) with labeled areas. The areas are given as follows: from t = 1 to t = 2, the area is 8 (positive, above the x-axis); from t = 2 to t = 3, the area is 5 (negative, below the x-axis); from t = 3 to t = 5, the area is 5 (positive, above the x-axis); and from t = 5 to t = 8, the area is 11 (negative, below the x-axis).
Step 3: Focus on the interval [2, 8]. Since F(8) involves the integral from t = 2 to t = 8, we only consider the areas between these bounds. Specifically, the areas are: from t = 2 to t = 3 (area = -5), from t = 3 to t = 5 (area = +5), and from t = 5 to t = 8 (area = -11).
Step 4: Compute the net area. Add the signed areas from t = 2 to t = 8: (-5) + (+5) + (-11). This sum represents the value of F(8).
Step 5: Conclude the process. The result of the sum in Step 4 gives the value of F(8). Ensure that the signs of the areas are correctly applied based on whether the region is above or below the x-axis.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integral
A definite integral represents the signed area under a curve between two points on the x-axis. It is calculated using the integral symbol and provides a numerical value that corresponds to the total area, accounting for areas above and below the x-axis. In the context of area functions, it helps determine the accumulated area from a starting point to a variable endpoint.
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Area Function
An area function, such as A(x) or F(x), is defined as the integral of a function from a fixed point to a variable upper limit. It quantifies the area under the curve of the function from the lower limit to x. This concept is crucial for evaluating specific values of the area function, as it allows us to compute the total area accumulated up to a certain point.
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Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus links differentiation and integration, stating that if F is an antiderivative of f on an interval, then the integral of f from a to b is equal to F(b) - F(a). This theorem is essential for evaluating area functions, as it allows us to compute definite integrals by finding the difference of the antiderivative values at the endpoints.
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