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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 5 - IntegrationProblema 5.2.34c
Capitolo 5, Problema 5.2.34c

{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n. 
(c) Calculate the left and right Riemann sums for the given value of n.


∫₀^π/2 cos 𝓍 d𝓍 ; n = 4

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Step 1: Understand the problem. You are tasked with approximating the definite integral ∫₀^(π/2) cos(𝓍) d𝓍 using left and right Riemann sums with n = 4 subintervals. Riemann sums approximate the area under a curve by summing the areas of rectangles.
Step 2: Divide the interval [0, π/2] into n = 4 subintervals. The width of each subinterval, Δ𝓍, is calculated as Δ𝓍 = (b - a) / n, where a = 0 and b = π/2. Substitute the values to find Δ𝓍.
Step 3: For the left Riemann sum, use the left endpoints of each subinterval to evaluate the function cos(𝓍). The left endpoints are x₀ = 0, x₁ = Δ𝓍, x₂ = 2Δ𝓍, and x₃ = 3Δ𝓍. Compute the sum: Left Riemann Sum = Δ𝓍 * [cos(x₀) + cos(x₁) + cos(x₂) + cos(x₃)].
Step 4: For the right Riemann sum, use the right endpoints of each subinterval to evaluate the function cos(𝓍). The right endpoints are x₁ = Δ𝓍, x₂ = 2Δ𝓍, x₃ = 3Δ𝓍, and x₄ = 4Δ𝓍. Compute the sum: Right Riemann Sum = Δ𝓍 * [cos(x₁) + cos(x₂) + cos(x₃) + cos(x₄)].
Step 5: Compare the left and right Riemann sums to understand how the choice of endpoints affects the approximation. These sums provide an estimate of the integral ∫₀^(π/2) cos(𝓍) d𝓍.

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Riemann Sums

Riemann sums are a method for approximating the value of a definite integral by dividing the area under a curve into smaller rectangles. The sum of the areas of these rectangles provides an estimate of the integral's value. Depending on whether the left or right endpoints of the subintervals are used, the sums can yield different approximations, which converge to the actual integral as the number of rectangles increases.
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Introduction to Riemann Sums

Definite Integral

A definite integral represents the signed area under a curve between two specified limits, often denoted as ∫_a^b f(x) dx. It quantifies the accumulation of quantities, such as area, over an interval [a, b]. The Fundamental Theorem of Calculus connects differentiation and integration, stating that the definite integral can be evaluated using the antiderivative of the function.
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Definition of the Definite Integral

Subintervals and n

In the context of Riemann sums, 'n' refers to the number of subintervals into which the interval [0, π/2] is divided. Each subinterval has a width of Δx = (b - a)/n, which determines the height of the rectangles used in the approximation. A larger value of n results in narrower subintervals, leading to a more accurate approximation of the definite integral.
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{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n. 

(c) Calculate the left and right Riemann sums for the given value of n.


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Approximating areas Estimate the area of the region bounded by the graph of ƒ(𝓍) = x² + 2 and the x-axis on [0, 2] in the following ways.

(c) Divide [0, 2] into n = 4 subintervals and approximate the area of the region using a right Riemann sum. Illustrate the solution geometrically.

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