Textbook Question
Evaluate the integrals in Exercises 47–68.
∫₀^π/3 sec² θ dθ
Ch. 5 - Integrals
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 5, Problem 5.PE.11
Area
In Exercises 11–14, find the total area of the region between the graph of ƒ and the x-axis.
ƒ(x) = x² - 4x + 3, 0 ≤ x ≤ 3
Verified step by step guidance1
Identify the function given: \(f(x) = x^{2} - 4x + 3\) and the interval \(0 \leq x \leq 3\) over which we want to find the total area between the graph and the x-axis.
Find the points where the graph intersects the x-axis by solving \(f(x) = 0\). This means solving the quadratic equation \(x^{2} - 4x + 3 = 0\).
Determine the sign of \(f(x)\) on each subinterval defined by the roots found in step 2 within the interval \([0,3]\). This helps to know where the graph is above or below the x-axis.
Set up the integral(s) for the total area. For regions where \(f(x)\) is positive, integrate \(f(x)\); for regions where \(f(x)\) is negative, integrate \(-f(x)\) to ensure the area is positive. The total area is the sum of these integrals.
Write the total area as the sum of definite integrals over the subintervals, for example: \(\int_{a}^{b} |f(x)| \, dx = \int_{a}^{c} f(x) \, dx - \int_{c}^{b} f(x) \, dx\) if \(f(x)\) changes sign at \(x=c\). Then, prepare to evaluate these integrals.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integral and Area Under a Curve
The definite integral of a function over an interval gives the net area between the graph and the x-axis. When the function is above the x-axis, the integral is positive; when below, it is negative. To find total area, consider absolute values of areas where the function dips below the axis.
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Definition of the Definite Integral
Finding Roots of the Function
Roots are points where the function crosses the x-axis, i.e., where f(x) = 0. Identifying these points within the interval helps split the region into subintervals where the function maintains a consistent sign, which is essential for calculating total area correctly.
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Evaluating Quadratic Functions
Understanding the shape and behavior of quadratic functions like f(x) = x² - 4x + 3 helps predict where the graph lies relative to the x-axis. This knowledge aids in determining intervals of positivity or negativity, crucial for setting up integrals to find areas.
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Related Practice
Textbook Question
Find the areas of the regions enclosed by the curves and lines in Exercises 15–26.
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Textbook Question
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Textbook Question
Definite Integrals
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Textbook Question
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Evaluate the integrals in Exercises 47–68.
∫₁² 4 dv
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