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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 7 - Transcendental FunctionsProblema 7.3.135
Capitolo 7, Problema 7.3.135

135. Find the area of the “triangular” region in the first quadrant that is bounded above by the curve y = e^(2x), below by the curve y = e^x, and on the right by the line x = ln(3).

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Identify the region bounded by the curves and the line: the area lies between y = e^{2x} (upper curve) and y = e^{x} (lower curve), from x = 0 (since it's in the first quadrant) to x = \(\ln\)(3) (right boundary).
Set up the integral for the area by subtracting the lower function from the upper function over the interval: the area A is given by the integral \( A = \int_{0}^{\ln(3)} \left(e^{2x} - e^{x}\right) \, dx \).
Recall the integral formulas for exponential functions: \( \int e^{ax} \, dx = \frac{1}{a} e^{ax} + C \), where a is a constant.
Integrate each term separately: compute \( \int e^{2x} \, dx = \frac{1}{2} e^{2x} \) and \( \int e^{x} \, dx = e^{x} \).
Evaluate the definite integral by substituting the limits x = 0 and x = \(\ln\)(3) into the integrated expression \( \left[ \frac{1}{2} e^{2x} - e^{x} \right]_{0}^{\ln(3)} \) and then find the difference to express the area.

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Definite Integrals for Area Calculation

Definite integrals are used to find the area under a curve between two points. When calculating the area between two curves, the integral of the difference of the functions over the given interval gives the enclosed area.
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Definition of the Definite Integral

Exponential Functions and Their Properties

Exponential functions like y = e^x and y = e^(2x) grow at rates proportional to their current value. Understanding their behavior and how to manipulate their expressions is essential for setting up the integral limits and integrand correctly.
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Properties of Functions

Determining Intersection Points and Boundaries

Identifying the region's boundaries involves finding where curves intersect and the limits of integration. Here, the vertical boundary x = ln(3) and the first quadrant restriction define the integration interval and ensure the area is correctly bounded.
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Determining Different Coordinates for the Same Point
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