Textbook Question
11-14. {Use of Tech} Compute the absolute and relative errors in using c to approximate x.
12. x = √2; c = 1.414
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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 8.8.6
5-8. Compute the following estimates of ∫(0 to 8) f(x) dx using the graph in the figure.
6. T(4)
Verified step by step guidance1
Identify that T(4) refers to the Trapezoidal Rule approximation of the integral using 4 subintervals over the interval [0, 8].
Calculate the width of each subinterval, \( \Delta x = \frac{8 - 0}{4} = 2 \).
Determine the x-values at the endpoints of the subintervals: 0, 2, 4, 6, and 8.
From the graph, find the corresponding function values \( f(x) \) at these points: \( f(0), f(2), f(4), f(6), f(8) \).
Apply the Trapezoidal Rule formula:
\[ T(4) = \frac{\Delta x}{2} \left[ f(0) + 2f(2) + 2f(4) + 2f(6) + f(8) \right] \]
This will give the estimate of the integral \( \int_0^8 f(x) \, dx \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integral as Area Under a Curve
The definite integral of a function over an interval represents the net area between the function's graph and the x-axis. It can be approximated by summing areas of shapes like rectangles or trapezoids under the curve, which is essential for estimating integrals from graphs.
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Definition of the Definite Integral
Trapezoidal Rule
The trapezoidal rule approximates the integral by dividing the interval into subintervals and approximating the area under the curve as trapezoids rather than rectangles. The formula averages the function values at the endpoints of each subinterval, providing a more accurate estimate than simple rectangular sums.
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5:50Power Rules
Partitioning the Interval
To apply numerical integration methods like the trapezoidal rule, the interval of integration is divided into equal or specified subintervals. For T(4), the interval [0,8] is divided into 4 subintervals of length 2, and function values at these points are used to calculate the trapezoidal sum.
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Related Practice
Textbook Question
"Electric field due to a line of charge A total charge of Q is distributed uniformly on a line segment of length 2L along the y-axis (see figure). The x-component of the electric field at a point (a, 0) is given by
Eₓ(a) = (kQa/2L) ∫-L L dy/(a² + y²)^(3/2),
where k is a physical constant and a > 0.
a. Confirm that Eₓ(a)=kQ / a √(a²+L²)
b. Letting ρ=Q / 2 L be the charge density on the line segment, show that if L → ∞, then Eₓ(a) = 2kρ / a.
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Textbook Question
60–69. Completing the square Evaluate the following integrals.
62. ∫ du / (2u² - 12u + 36)
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Textbook Question
29-34. {Use of Tech} Comparing the Midpoint and Trapezoid Rules
Apply the Midpoint and Trapezoid Rules to the following integrals. Make a table similar to Table 8.5 showing the approximations and errors for n = 4, 8, 16, and 32. The exact values of the integrals are given for computing the error.
30. ∫(0 to 6) (x³/16 - x) dx = 4
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Textbook Question
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
10. ∫ (x³ + 3x² + 1)/(x³ + 1) dx
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Textbook Question
108. Arc length Find the length of the curve y = ln(x) on the interval [1, e^2].
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