Area Find (i) the net area and (ii) the area of the following regions. Graph the function and indicate the region in question.

The region bounded by y = 6 cos 𝓍 and the 𝓍-axis between 𝓍 = ―π/2 and 𝓍 = π

검증된 단계별 안내

Step 1: Understand the problem. You are tasked with finding (i) the net area and (ii) the total area of the region bounded by the function y = 6 cos(𝓍) and the 𝓍-axis between 𝓍 = -π/2 and 𝓍 = π. The net area accounts for signed areas (positive above the 𝓍-axis and negative below), while the total area considers only the magnitude of the regions.

Step 2: Graph the function y = 6 cos(𝓍) over the interval [-π/2, π]. The cosine function oscillates between -1 and 1, and multiplying by 6 scales the amplitude to [-6, 6]. Identify where the function crosses the 𝓍-axis (i.e., where y = 0) within the interval. These points divide the region into subintervals where the function is either positive or negative.

Step 3: Set up the integral for the net area. The net area is calculated using the definite integral of y = 6 cos(𝓍) over the interval [-π/2, π]. Use the formula:

\int a_{- π / 2} b_{π} 6 \cos π

Step 4: Set up the integral for the total area. To find the total area, you need to integrate the absolute value of y = 6 cos(𝓍) over the interval [-π/2, π]. This requires splitting the integral into subintervals where the function is positive and negative, and then taking the absolute value of the function in the negative subinterval. Use the formula:

\int a_{- π / 2} b_{π} 6 \cos π

Step 5: Solve the integrals. For the net area, evaluate the definite integral of 6 cos(𝓍) over [-π/2, π]. For the total area, split the integral into subintervals where the function is positive and negative, and evaluate the absolute value of the function in the negative subinterval. Combine the results to find the total area. Ensure proper use of trigonometric identities and integration rules.

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주요 개념

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

Definite Integral

The definite integral is a fundamental concept in calculus that represents the net area under a curve between two specified limits. It is calculated using the integral symbol and provides a way to quantify the accumulation of quantities, such as area, over an interval. In this context, it will be used to find the area between the curve y = 6 cos x and the x-axis from x = -π/2 to x = π.

Net Area vs. Total Area

Net area refers to the signed area between a curve and the x-axis, where areas above the x-axis are positive and those below are negative. In contrast, total area considers all areas as positive, regardless of their position relative to the x-axis. Understanding this distinction is crucial for accurately calculating the areas in the given problem, especially since the cosine function oscillates above and below the x-axis.

Graphing Trigonometric Functions

Graphing trigonometric functions, such as y = 6 cos x, involves plotting the function's values over a specified interval to visualize its behavior. This includes identifying key features like amplitude, period, and intercepts. For this problem, graphing the function will help in understanding the regions bounded by the curve and the x-axis, which is essential for determining both the net and total areas.

Area Find (i) the net area and (ii) the area of the following regions. Graph the function and indicate the region in question.The region bounded by y = 6 cos 𝓍 and the 𝓍-axis between 𝓍 = ―π/2 and 𝓍 = π