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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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.

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.