3. Estimation Error. The definite integral below can be computed using the Fundamental Theorem of Calculus. Calculate the actual value and then fill in the table with the given estimates and errors. What is the relationship between the number of rectangles and the size of the error? $\int_1^{\sqrt{5}} \frac{1}{x^2 + 1} dx$ Exact value = $[\tan^{-1}(x)]^{\sqrt{5}}_1 = $ In the table use the midpoint estimate with $n$ rectangles, $M_n$, and use Err = |actual - estimate| for the error column. \begin{tabular}{|c|c|c|} \hline n & $M_n$ & Err \\ \hline 1 & & \\ 10 & & \\ 100 & & \\ 1000 & & \\ \hline \end{tabular} What is the relationship between the number of rectangles and the size of the error? Hint: It helps to write the error in scientific notation.
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Since the problem statement is not clear about the function and the interval, let's assume we are working with the following integral: $$\int_a^b f(x) dx$$ Now, we can use the Fundamental Theorem of Calculus to find the exact value: $$F(b) - F(a)$$ where F(x) Show more…
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(c) Use the Fundamental Theorem of Calculus to compute the exact value of I = ∫₀³ x² dx. I = (d) We define the error in an approximation of a definite integral to be the difference between the integral's exact value and the approximation's value. What is the error that results from using L₃? From R₃? From M₃? I − L₃ = I − R₃ = I − M₃ = (e) Let us develop a new approach to estimating the value of a definite integral known as the Trapezoid Rule. The basic idea is to use trapezoids, rather than rectangles, to estimate the area under a curve. What is the formula for the area of a trapezoid with bases of length b and B and height h? Area of Trapeziod = (f) Working by hand, estimate the area under f(x) = x² on [0, 3] using three subintervals and three corresponding trapezoids. Call this value T₃. T₃ = What is the error in this approximation? I − T₃ = How does it compare to the errors you calculated in (d)?
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Use Trapezoid approximation with n = 4 equal subintervals to approximate the value of ∫[0,1] 10e^(x^2) dx. Find an upper bound for the error in the Trapezoid approximation above. (Use this theorem.) Step 1 of 6 Use Trapezoid approximation with n = 4 equal subintervals to approximate the value of ∫[0,1] 10e^(x^2) dx. To use trapezoid approximation to approximate the integral of the function f(x) = 10e^(x^2) on the interval [0, 1] with n = 4, divide the interval [0, 1] into 4 subintervals of equal width. Calculate Δx and use it to fill in the table below. Δx = (b-a)/n =
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