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 guidance

1. 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}].
2. 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 \( A_2 = \int_0^{\frac{\pi}{6}} \sec(x) \, dx \).
3. Recall the antiderivatives needed: The integral of \( \tan(x) \) is \( -\ln|\cos(x)| + C \), and the integral of \( \sec(x) \) is \( \ln|\sec(x) + \tan(x)| + C \).
4. 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 \( A_2 \).
5. Compare the two computed areas \( A_1 \) and \( A_2 \) to determine which region has the greater area.

Key Concepts

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.

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.

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.