Textbook Question
Area of regions Compute the area of the region bounded by the graph of ƒ and the 𝓍-axis on the given interval. You may find it useful to sketch the region.
ƒ(𝓍) = 2 sin 𝓍/4 on [0, 2π]
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9. Graphical Applications of Integrals
Area Between Curves
1:42 minutesProblem 5.3.14a
Textbook Question
Area functions The graph of ƒ is shown in the figure. Let A(x) = ∫₀ˣ ƒ(t) dt and F(x) = ∫₂ˣ ƒ(t) dt be two area functions for ƒ. Evaluate the following area functions.
(a) A(2)
Verified step by step guidance1
Step 1: Understand the problem. We are tasked with evaluating A(2), where A(x) = ∫₀ˣ ƒ(t) dt. This represents the area under the curve of ƒ(t) from t = 0 to t = x.
Step 2: Analyze the graph. The graph shows the function ƒ(t) and the areas of specific regions. From t = 0 to t = 1, the area is shaded green and labeled as 'Area = 8'. This area is above the t-axis, so it contributes positively to the integral.
Step 3: From t = 1 to t = 2, the graph dips below the t-axis, and the area is shaded green and labeled as 'Area = 5'. Since this region is below the t-axis, it contributes negatively to the integral.
Step 4: To compute A(2), sum the contributions of the areas from t = 0 to t = 2. Specifically, add the positive area from t = 0 to t = 1 and subtract the negative area from t = 1 to t = 2.
Step 5: Write the expression for A(2): A(2) = (Area from t = 0 to t = 1) - (Area from t = 1 to t = 2). Substitute the values from the graph: A(2) = 8 - 5.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integral
A definite integral represents the signed area under a curve between two points on the x-axis. It is calculated using the integral symbol and limits of integration, providing a numerical value that corresponds to the net area, accounting for areas above and below the x-axis.
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Area Function
An area function, such as A(x) = ∫₀ˣ ƒ(t) dt, defines the accumulated area under the curve of a function ƒ from a starting point to a variable endpoint x. This function helps in understanding how the area changes as x varies, and it is crucial for evaluating specific area values at given points.
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Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus links differentiation and integration, stating that if F is an antiderivative of f on an interval, then the definite integral of f from a to b can be computed as F(b) - F(a). This theorem is essential for evaluating area functions and understanding their properties.
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