11. The function f is continuous on the closed interval [2, 8] and has values that are given in the table above. Using the subintervals [2 , 5], [5, 7], and [7, 8], what is the trapezoidal approximation of ∫ f(x) dx
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It's a method used to estimate the definite integral of a function. The area under the curve of the function is divided into trapezoids, and the sum of the areas of these trapezoids gives the approximation of the integral. The formula for the area of a trapezoid Show more…
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1. Use the trapezoidal rule to approximate integral from 0 to 2 of x^3 dx for n = 4. 2. The function f is continuous on [2, 8] and has values that are given in the table below. Using the subintervals [2, 5], [5, 7], and [7, 8], what is the trapezoidal approximation of integral from 2 to 8 of f(x) dx? x | 2 | 5 | 7 | 8 f(x) | 10 | 30 | 40 | 20
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Trapezoidal Rule In Exercises 27 and $28,$ use the Trapezoidal Rule with $n=8$ to approximate the definite integral. Compare the result with the exact value and the approximation obtained with $n=8$ and the Midpoint Rule. Which approximation technique appears to be better? Let $f$ be continuous on $[a, b]$ and let $n$ be the number of equal sub intervals (see figure). Then the Trapezoidal Rule for approximating $\int_{a}^{b} f(x) d x$ is $$ \frac{b-a}{2 n}\left[f\left(x_{0}\right)+2 f\left(x_{1}\right)+\cdots+2 f\left(x_{n-1}\right)+f\left(x_{n}\right)\right] $$ $$ \int_{0}^{2} x^{3} d x $$
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