Textbook Question
Area of regions Compute the area of the region bounded by the graph of Ζ and the π-axis on the given interval. You may find it useful to sketch the region.
Ζ(π) = 16βπΒ² on [β4, 4]
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9. Graphical Applications of Integrals
Area Between Curves
1:59 minutesProblem 5.3.4
Textbook Question
Let Ζ(π) = c, where c is a positive constant. Explain why an area function of Ζ is an increasing function.
Verified step by step guidance1
Step 1: Recognize that the function Ζ(π) = c is a constant function, where c is a positive constant. This means that for any value of π, Ζ(π) always equals c.
Step 2: Recall the concept of an area function. An area function represents the accumulated area under the curve of Ζ(π) from a starting point to a variable endpoint. Mathematically, this is expressed as A(π) = β«[a,π] Ζ(t) dt, where a is the starting point and t is the variable of integration.
Step 3: Substitute Ζ(t) = c into the integral. The area function becomes A(π) = β«[a,π] c dt. Since c is constant, it can be factored out of the integral, resulting in A(π) = c β«[a,π] dt.
Step 4: Evaluate the integral β«[a,π] dt. The integral of 1 with respect to t is t, so β«[a,π] dt = [t] evaluated from a to π, which simplifies to π - a. Therefore, A(π) = c(π - a).
Step 5: Observe that A(π) = c(π - a) is a linear function of π with a positive slope (since c > 0). As π increases, A(π) also increases, which confirms that the area function is an increasing function.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Constant Function
A constant function is a function that always returns the same value, regardless of the input. In this case, Ζ(π) = c means that for any value of π, the output is the constant c. This characteristic implies that the graph of the function is a horizontal line, which is crucial for understanding how the area under the curve behaves.
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Area Under the Curve
The area under the curve of a function represents the integral of that function over a specified interval. For a constant function like Ζ(π) = c, the area can be calculated as the product of the constant value and the width of the interval. As the interval increases, the area also increases linearly, which is key to understanding why the area function is increasing.
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Increasing Function
An increasing function is one where, as the input value increases, the output value also increases. In the context of the area function derived from a constant function, since the area grows as the interval expands, the area function is classified as increasing. This means that for any two intervals where the second is larger than the first, the area calculated for the second will always be greater.
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