Determine the area of the shaded region in the following figures.

(Hint: Find the intersection point by inspection.)

검증된 단계별 안내

Step 1: Identify the equations of the curves that bound the shaded region. The curves are y = 2^x (an exponential function) and y = 3 - x (a linear function).

Step 2: Find the intersection point of the two curves by solving the equation 2^x = 3 - x. This can be done by inspection or algebraically. The intersection point will provide the limits of integration.

Step 3: Set up the integral to calculate the area of the shaded region. The area is given by the integral of the upper curve minus the lower curve over the interval determined by the intersection points. The integral is: ∫[x1 to x2] ((3 - x) - (2^x)) dx, where x1 and x2 are the x-coordinates of the intersection points.

Step 4: Break down the integral into manageable parts. For example, split the integral into ∫[x1 to x2] (3 dx) - ∫[x1 to x2] (x dx) - ∫[x1 to x2] (2^x dx). Each term can be evaluated separately using standard integration techniques.

Step 5: Evaluate each integral. For ∫(3 dx), use the formula for the integral of a constant. For ∫(x dx), use the power rule. For ∫(2^x dx), use the formula for the integral of an exponential function. Combine the results to find the total area.

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주요 개념

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

Intersection of Functions

The intersection of functions occurs where their graphs meet, meaning they have the same output for a given input. In this problem, finding the intersection points of the curves y = 2^x and y = 3 - x is essential, as these points define the boundaries of the shaded area whose area we need to calculate.

Area Under a Curve

The area under a curve can be calculated using definite integrals, which represent the accumulation of quantities. In this case, the area of the shaded region between the two curves can be found by integrating the difference between the upper function (3 - x) and the lower function (2^x) over the interval defined by their intersection points.

Definite Integral

A definite integral calculates the net area under a curve between two specified limits. It is denoted as ∫[a, b] f(x) dx, where f(x) is the function being integrated, and a and b are the limits of integration. This concept is crucial for determining the area of the shaded region in the given problem, as it provides a systematic way to compute the area between the two intersecting curves.

Determine the area of the shaded region in the following figures.(Hint: Find the intersection point by inspection.)