42. Approximating integrals The function f is twice differentiable on (-∞, ∞). Values of f at various points on [0, 20] are given in the table.

a. Approximate ∫(0 to 120) f(x) dx in three way using a left Riemann sum, a right Riemann sum and the Trapezoid Rule

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Step 1: To approximate the integral ∫(0 to 20) f(x) dx, we will use three methods: the Left Riemann Sum, the Right Riemann Sum, and the Trapezoid Rule. First, note the given data points for x and f(x) from the table.

Step 2: For the Left Riemann Sum, use the formula: L = Σ f(x_i) * Δx, where x_i are the left endpoints of each subinterval. Calculate Δx for each subinterval (e.g., Δx = x_(i+1) - x_i) and multiply by the corresponding f(x) values at the left endpoints.

Step 3: For the Right Riemann Sum, use the formula: R = Σ f(x_(i+1)) * Δx, where x_(i+1) are the right endpoints of each subinterval. Again, calculate Δx for each subinterval and multiply by the corresponding f(x) values at the right endpoints.

Step 4: For the Trapezoid Rule, use the formula: T = (1/2) * Σ [f(x_i) + f(x_(i+1))] * Δx. Here, calculate the average of f(x) values at each pair of endpoints and multiply by the width of the subinterval Δx.

Step 5: Add up the contributions from all subintervals for each method (Left Riemann Sum, Right Riemann Sum, and Trapezoid Rule) to approximate the integral. Ensure that you use the correct Δx for each subinterval as given in the table.

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Riemann Sums

Riemann sums are a method for approximating the definite integral of a function over an interval. They involve partitioning the interval into subintervals and summing the areas of rectangles formed by evaluating the function at specific points, such as the left endpoint, right endpoint, or midpoint of each subinterval. This technique provides a way to estimate the area under the curve, which is the essence of integration.

Trapezoid Rule

The Trapezoid Rule is a numerical method for estimating the definite integral of a function by approximating the area under the curve as a series of trapezoids rather than rectangles. It calculates the area of each trapezoid formed between two points on the x-axis, using the average of the function values at these points. This method generally provides a more accurate approximation than Riemann sums, especially for functions that are linear or nearly linear over small intervals.

Definite Integral

A definite integral represents the signed area under a curve defined by a function over a specific interval [a, b]. It is calculated as the limit of Riemann sums as the number of subintervals approaches infinity. The definite integral not only provides a numerical value representing the area but also has applications in various fields, such as physics and engineering, where it can represent quantities like distance, area, and volume.

42. Approximating integrals The function f is twice differentiable on (-∞, ∞). Values of f at various points on [0, 20] are given in the table.a. Approximate ∫(0 to 120) f(x) dx in three way using a left Riemann sum, a right Riemann sum and the Trapezoid Rule