Multiple Choice
Find the area enclosed by one loop of the curve .
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9. Graphical Applications of Integrals
Area Between Curves
2:29 minutesProblem 5.3.71
Textbook Question
Areas of regions Find the area of the region bounded by the graph of Ζ and the π-axis on the given interval.
Ζ(π) = sin π on [βΟ/4, 3Ο/4]
Verified step by step guidance1
Step 1: Understand the problem. We are tasked with finding the area of the region bounded by the graph of Ζ(π) = sin(π) and the π-axis over the interval [βΟ/4, 3Ο/4]. This involves integrating the function Ζ(π) = sin(π) over the given interval.
Step 2: Set up the definite integral. The area can be calculated using the formula:
Step 3: Evaluate the integral of sin(π). Recall that the integral of sin(π) is βcos(π). Substitute this into the integral:
Step 4: Apply the Fundamental Theorem of Calculus. Evaluate βcos(π) at the bounds of the interval [βΟ/4, 3Ο/4]. This means substituting the upper bound (3Ο/4) and the lower bound (βΟ/4) into βcos(π) and subtracting the results.
Step 5: Simplify the expression. Compute the values of βcos(3Ο/4) and βcos(βΟ/4), then subtract them to find the area. Remember that the area is always positive, so take the absolute value if necessary.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integral
The definite integral of a function over a specified interval represents the net area between the graph of the function and the x-axis. It is calculated using the Fundamental Theorem of Calculus, which connects differentiation and integration. In this context, the area can be found by integrating the function Ζ(π) = sin π from the lower limit of -Ο/4 to the upper limit of 3Ο/4.
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Area Under the Curve
The area under the curve of a function can be positive or negative depending on whether the function is above or below the x-axis. When calculating the total area, it is important to consider the absolute value of the areas where the function is negative, as these contribute to the overall area of the region bounded by the graph and the x-axis.
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Properties of the Sine Function
The sine function, Ζ(π) = sin π, is periodic and oscillates between -1 and 1. Understanding its behavior over the interval [-Ο/4, 3Ο/4] is crucial for determining the area, as it crosses the x-axis at specific points. This knowledge helps in identifying the segments of the interval where the function is positive or negative, which is essential for accurate area calculation.
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