Textbook Question
3. Explain geometrically how the Trapezoid Rule is used to approximate a definite integral.
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8. Definite Integrals
Riemann Sums
2:39 minutesProblem 8.8.15
Textbook Question
15-18. {Use of Tech} Midpoint Rule approximations. Find the indicated Midpoint Rule approximations to the following integrals.
15. ∫(2 to 10) 2x² dx using n = 1, 2, and 4 subintervals
Verified step by step guidance1
Step 1: Understand the Midpoint Rule. The Midpoint Rule is a numerical method to approximate the value of a definite integral. It divides the interval into 'n' subintervals, calculates the midpoint of each subinterval, evaluates the function at these midpoints, and then multiplies by the width of the subintervals.
Step 2: Identify the integral and interval. The integral is ∫(2 to 10) 2x² dx, and the interval is [2, 10]. The number of subintervals (n) will vary as specified: n = 1, n = 2, and n = 4.
Step 3: Calculate the width of each subinterval (Δx). The formula for Δx is Δx = (b - a) / n, where 'a' is the lower limit (2), 'b' is the upper limit (10), and 'n' is the number of subintervals. Compute Δx for n = 1, n = 2, and n = 4.
Step 4: Determine the midpoints of each subinterval. For each subinterval, the midpoint is calculated as (x_i + x_(i+1)) / 2, where x_i and x_(i+1) are the endpoints of the subinterval. Compute the midpoints for n = 1, n = 2, and n = 4.
Step 5: Apply the Midpoint Rule formula. The approximation is given by M_n = Δx * Σ[f(midpoint_i)], where f(x) = 2x² is the function being integrated, Δx is the width of the subintervals, and midpoint_i are the midpoints of the subintervals. Evaluate the sum for n = 1, n = 2, and n = 4.
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2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Midpoint Rule
The Midpoint Rule is a numerical method used to approximate the value of a definite integral. It involves dividing the interval of integration into subintervals, calculating the midpoint of each subinterval, and then evaluating the function at these midpoints. The area under the curve is then approximated by summing the areas of rectangles formed by these midpoints, multiplied by the width of the subintervals.
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Definite Integral
A definite integral represents the signed area under a curve defined by a function over a specific interval [a, b]. It is denoted as ∫(a to b) f(x) dx and provides a way to calculate total accumulation, such as area, volume, or other quantities. The Fundamental Theorem of Calculus connects the concept of differentiation with integration, allowing for the evaluation of definite integrals using antiderivatives.
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Subintervals
Subintervals are smaller segments into which the main interval of integration is divided when applying numerical methods like the Midpoint Rule. The number of subintervals, denoted as 'n', affects the accuracy of the approximation; more subintervals generally lead to a better approximation of the integral. Each subinterval has a width calculated as (b - a) / n, where [a, b] is the interval of integration.
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