Textbook Question
91. [Use of Tech] Regions bounded by exponentials Let a > 0 and let R be the region bounded by the graph of y = e^(-a·x) and the x-axis
on the interval [b, ∞).
a. Find A(a,b), the area of R as a function of a and b.
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Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 8, Problem 8.7.86a
Arc length of a parabola Let L(c) be the length of the parabola f(x) = x² from x = 0 to x = c, where c ≥ 0 is a constant.
a. Find an expression for L.
Verified step by step guidance1
Step 1: Recall the formula for the arc length of a curve y = f(x) from x = a to x = b: \( L = \int_a^b \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \). This formula will be used to find the arc length of the parabola.
Step 2: Identify the function \( f(x) = x^2 \) and compute its derivative \( \frac{dy}{dx} = \frac{d}{dx}(x^2) = 2x \). Substitute \( \frac{dy}{dx} \) into the arc length formula.
Step 3: Substitute \( \frac{dy}{dx} = 2x \) into the formula \( L = \int_0^c \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \), resulting in \( L = \int_0^c \sqrt{1 + (2x)^2} \, dx \). Simplify the expression inside the square root.
Step 4: Simplify \( \sqrt{1 + (2x)^2} \) to \( \sqrt{1 + 4x^2} \). The arc length formula now becomes \( L = \int_0^c \sqrt{1 + 4x^2} \, dx \). This is the expression for the arc length.
Step 5: Note that the integral \( \int_0^c \sqrt{1 + 4x^2} \, dx \) may require advanced techniques such as substitution or numerical methods to evaluate for specific values of \( c \). The expression \( L = \int_0^c \sqrt{1 + 4x^2} \, dx \) is the general formula for the arc length of the parabola.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Arc Length Formula
The arc length of a curve defined by a function y = f(x) from x = a to x = b is given by the formula L = ∫[a to b] √(1 + (dy/dx)²) dx. This formula derives from the Pythagorean theorem, where the infinitesimal segments of the curve are approximated as straight lines.
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Arc Length of Parametric Curves
Derivative of a Function
The derivative of a function, denoted as dy/dx, represents the rate of change of the function with respect to x. For the parabola f(x) = x², the derivative is f'(x) = 2x, which is essential for calculating the arc length as it appears in the arc length formula.
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06:30Derivatives of Other Trig Functions
Definite Integral
A definite integral calculates the accumulation of quantities, such as area under a curve, over a specific interval [a, b]. In the context of finding the arc length, the definite integral is used to sum the lengths of infinitesimal segments of the curve from x = 0 to x = c, providing the total length L(c).
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Related Practice
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. ∫(3/(x² + 4)) dx = ∫(3/x²) dx + ∫(3/4) dx.
Textbook Question
60. Two Methods
a. Evaluate ∫(x · ln(x²)) dx using the substitution u = x² and evaluating ∫(ln(u)) du.
Textbook Question
57. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. If x = 4 tanθ, then cscθ = 4/x.
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. Suppose ∫_a^b f(x) dx is approximated with Simpson’s Rule using n = 18 subintervals, where |f^(4)(x)| ≤ 1 on [a, b]. The absolute error E_S in approximating the integral satisfies E_S ≤ (Δx)^5 / 10.
Textbook Question
Gamma function The gamma function is defined by Γ(p) = ∫ from 0 to ∞ of x^(p-1) e^(-x) dx, for p not equal to zero or a negative integer.
a. Use the reduction formula ∫ from 0 to ∞ of x^p e^(-x) dx = p ∫ from 0 to ∞ of x^(p-1) e^(-x) dx for p = 1, 2, 3, ...
to show that Γ(p + 1) = p! (p factorial).
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