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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 8 - Integration TechniquesProblem 8.8.46c
Chapter 8, Problem 8.8.46c

45–48. {Use of Tech} Trapezoid Rule and Simpson’s Rule Consider the following integrals and the given values of n.
46. ∫(0 to 2) x⁴ dx; n = 30
c. Compute the absolute errors in the Trapezoid Rule and Simpson’s Rule with 2n subintervals.

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Step 1: Understand the problem. You are tasked with calculating the absolute errors in the Trapezoid Rule and Simpson’s Rule for the integral ∫(0 to 2) x⁴ dx using 2n subintervals, where n = 30. This means you will use 60 subintervals for both methods.
Step 2: Recall the formulas for the Trapezoid Rule and Simpson’s Rule. The Trapezoid Rule approximates the integral as a sum of trapezoidal areas, while Simpson’s Rule uses parabolic segments for approximation. Both methods depend on the number of subintervals and the function being integrated.
Step 3: Compute the exact value of the integral ∫(0 to 2) x⁴ dx. To do this, use the power rule for integration: ∫x⁴ dx = (x⁵)/5 + C. Evaluate this definite integral from 0 to 2 to find the exact value. This will be used to calculate the absolute errors later.
Step 4: Apply the Trapezoid Rule and Simpson’s Rule with 60 subintervals. For the Trapezoid Rule, divide the interval [0, 2] into 60 equal parts, calculate the function values at each endpoint, and use the formula for the Trapezoid Rule. For Simpson’s Rule, divide the interval into 60 subintervals, calculate the function values at endpoints and midpoints, and apply Simpson’s Rule formula.
Step 5: Calculate the absolute errors. The absolute error for each method is the absolute difference between the exact value of the integral and the approximate value obtained using the respective rule. Use the exact value from Step 3 and the approximations from Step 4 to compute these errors.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Trapezoid Rule

The Trapezoid Rule is a numerical method for approximating the definite integral of a function. It works by dividing the area under the curve into trapezoids rather than rectangles, which provides a better approximation. The formula involves calculating the average of the function values at the endpoints of each subinterval and multiplying by the width of the subintervals. This method is particularly useful when an analytical solution is difficult to obtain.
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Power Rules

Simpson’s Rule

Simpson’s Rule is another numerical technique for estimating the value of a definite integral. It approximates the integrand by a quadratic polynomial, using parabolic segments instead of straight lines. The rule requires an even number of subintervals and combines the function values at the endpoints and midpoints of the intervals. This method generally provides a more accurate approximation than the Trapezoid Rule, especially for smooth functions.
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Absolute Error

Absolute error measures the difference between the exact value of an integral and the approximate value obtained using numerical methods like the Trapezoid Rule or Simpson’s Rule. It is calculated as the absolute value of the difference between these two values. Understanding absolute error is crucial for assessing the accuracy of numerical approximations and helps in determining how close the approximation is to the true value.
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Determining Error and Relative Error
Related Practice
Textbook Question

109. Escape velocity and black holes The work required to launch an object from the surface of Earth to outer space is given by W = ∫ from R to ∞ of F(x) dx, where R = 6370 km is the approximate radius of Earth, F(x) = (GMm)/x² is the gravitational force between Earth and the object, G is the gravitational constant, M is the mass of Earth, m is the mass of the object, and GM = 4 × 10¹⁴ m³/s².

c. The French scientist Laplace anticipated the existence of black holes in the 18th century with the following argument: If a body has an escape velocity that equals or exceeds the speed of light, c = 300,000 km/s, then light cannot escape the body and it cannot be seen. Show that such a body has a radius R ≤ 2GM/c². For Earth to be a black hole, what would its radius need to be?

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Textbook Question

58. Two Methods Evaluate ∫(from 0 to π/3) sin(x) · ln(cos(x)) dx in the following two ways:

b. Use substitution.

Textbook Question

60. Two Methods

c. Verify that your answers to parts (a) and (b) are consistent.

Textbook Question

94. [Use of Tech] Skydiving A skydiver has a downward velocity given by v(t) = V_T [(1 - e^(-2gt/V_T))/(1 + e^(-2gt/V_T))],

where t = 0 is the instant the skydiver starts falling, g = 9.8 m/s² is the acceleration due to gravity, and V_T is the terminal velocity of the skydiver.

c. Verify by integration that the position function is given by

s(t) = V_T t + (V_T²/g) ln[(1 + e^(-2gt/V_T))/2],

where s'(t) = v(t) and s(0) = 0.

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Textbook Question

75. {Use of Tech} Oscillator displacements Suppose a mass on a spring that is slowed by friction has the position function:

s(t) = e⁻ᵗ sin t

c. Generalize part (b) and find the average value of the position on the interval [nπ, (n+1)π], for n = 0, 1, 2, ...

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, ∞).

c. Find the minimum value b* such that when b > b*, there exists some a > 0 where A(a,b) = 2.