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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 8.8.6
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.
Graph of a function with labeled axes, showing a fluctuating curve plotted on a grid.
6. T(4)

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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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Power 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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Interval of Convergence
Related Practice
Textbook Question

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