Textbook Question
35-38. Area and volume Let R be the region in the first quadrant bounded by the graph of
Find the area of the region R.
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9. Graphical Applications of Integrals
Area Between Curves
2:21 minutesProblem 8.R.122
Textbook Question
122. Comparing areas The region R₁ is bounded by the graph of y = tan(x) and the x-axis on the interval [0, π/3].
The region R₂ is bounded by the graph of y = sec(x) and the x-axis on the interval [0, π/6]. Which region has the greater area?
Verified step by step guidance1
Identify the two regions and their boundaries: Region R₁ is bounded by y = tan(x) and the x-axis on [0, \(\frac{\pi}{3}\)], and Region R₂ is bounded by y = sec(x) and the x-axis on [0, \(\frac{\pi}{6}\)].
Set up the definite integrals to find the areas of each region. The area of R₁ is given by the integral \( A_1 = \int_0^{\frac{\pi}{3}} \tan(x) \, dx \), and the area of R₂ is given by \( A_2 = \int_0^{\frac{\pi}{6}} \sec(x) \, dx \).
Recall the antiderivatives needed: The integral of \( \tan(x) \) is \( -\ln|\cos(x)| + C \), and the integral of \( \sec(x) \) is \( \ln|\sec(x) + \tan(x)| + C \).
Evaluate each definite integral by applying the Fundamental Theorem of Calculus, substituting the upper and lower limits into the antiderivatives for both \( A_1 \) and \( A_2 \).
Compare the two computed areas \( A_1 \) and \( A_2 \) to determine which region has the greater area.
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integrals and Area Under a Curve
The definite integral of a function over an interval represents the net area between the graph of the function and the x-axis. For positive functions, this integral gives the exact area of the region bounded by the curve and the axis. Calculating these integrals helps compare areas of different regions.
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Definition of the Definite Integral
Properties of the Functions tan(x) and sec(x)
The functions tan(x) and sec(x) are trigonometric functions with distinct behaviors on given intervals. Understanding their values and growth on [0, π/3] and [0, π/6] respectively is essential, as tan(x) increases from 0 to √3, while sec(x) increases from 1 to 2, affecting the area under their curves.
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Comparison of Areas via Integration Limits and Function Behavior
Comparing areas requires evaluating integrals over different intervals and functions. Since the intervals and functions differ, analyzing how the length of the interval and the magnitude of the function values influence the total area is crucial to determine which region has the greater area.
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