Ch. 5 - Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 5 - Integration

Problem 5.R.9d

Chapter 5, Problem 5.R.9d

Integration by Riemann sums Consider the integral ∫₁⁴ (3𝓍― 2) d𝓍.

(a) Evaluate the right Riemann sum for the integral with n = 3 .

Verified step by step guidance

Step 1: Understand the problem. The integral ∫₁⁴ (3𝓍 - 2) d𝓍 represents the area under the curve of the function f(𝓍) = 3𝓍 - 2 from 𝓍 = 1 to 𝓍 = 4. To approximate this integral using the right Riemann sum, we divide the interval [1, 4] into n = 3 subintervals.

Step 2: Determine the width of each subinterval. The width Δ𝓍 is calculated as Δ𝓍 = (b - a) / n, where a = 1, b = 4, and n = 3. Substitute these values into the formula to find Δ𝓍.

Step 3: Identify the right endpoints of each subinterval. The right endpoints are calculated as 𝓍₁ = a + Δ𝓍, 𝓍₂ = a + 2Δ𝓍, and 𝓍₃ = a + 3Δ𝓍. Use the value of Δ𝓍 from Step 2 to compute these endpoints.

Step 4: Evaluate the function f(𝓍) = 3𝓍 - 2 at each right endpoint. Substitute the values of 𝓍₁, 𝓍₂, and 𝓍₃ into the function to find f(𝓍₁), f(𝓍₂), and f(𝓍₃).

Step 5: Compute the right Riemann sum. Multiply each function value f(𝓍ᵢ) by the width Δ𝓍 and sum them up: Right Riemann Sum = Δ𝓍 × [f(𝓍₁) + f(𝓍₂) + f(𝓍₃)]. This gives the approximation for the integral.

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Key Concepts

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

Riemann Sums

Riemann sums are a method for approximating the value of a definite integral by dividing the area under a curve into small rectangles. The sum of the areas of these rectangles provides an estimate of the integral's value. Depending on the chosen points (left, right, or midpoint) for the height of the rectangles, different types of Riemann sums can be calculated, which converge to the actual integral as the number of rectangles increases.

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 ∫ₐᵇ f(x) dx and can be interpreted as the limit of Riemann sums as the number of subdivisions approaches infinity. The Fundamental Theorem of Calculus connects differentiation and integration, allowing us to evaluate definite integrals using antiderivatives.

Partitioning the Interval

Partitioning the interval involves dividing the range of integration into smaller subintervals, which is essential for calculating Riemann sums. For n subintervals, the width of each subinterval is Δx = (b - a)/n. In this case, with n = 3 for the integral from 1 to 4, the interval is divided into three equal parts, allowing for the evaluation of the function at specific points to approximate the area under the curve.

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Related Practice

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Area of regions Compute the area of the region bounded by the graph of ƒ and the 𝓍-axis on the given interval. You may find it useful to sketch the region.

ƒ(𝓍) = 16―𝓍² on [―4, 4]

105

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Find the average value of ƒ(𝓍) = e²ˣ on [0, ln 2] .

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Textbook Question

Properties of integrals Suppose ∫₁⁴ ƒ(𝓍) d𝓍 = 6 , ∫₁⁴ g(𝓍) d𝓍 = 4 and ∫₃⁴ ƒ(𝓍) d𝓍 = 2 . Evaluate the following integrals or state that there is not enough information.

―∫₄¹ 2ƒ(𝓍) d𝓍

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Evaluating integrals Evaluate the following integrals.

∫ (9𝓍⁸―7𝓍⁶) d𝓍

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Textbook Question

Area functions and the Fundamental Theorem Consider the function

ƒ(t) = { t if ―2 ≤ t < 0

t²/2 if 0 ≤ t ≤ 2

and its graph shown below. Let F(𝓍) = ∫₋₁ˣ ƒ(t) dt and G(𝓍) = ∫₋₂ˣ ƒ(t) dt.

(e) Evaluate F ''(―1) and F ''(1). Interpret these values.