Trapezoidal Rule gives exact value of the integral when the integrand is a2 pointsLinear functionQuadratic functionCubic functionPolynomial of any degree
Question
Trapezoidal Rule gives exact value of the integral when the integrand is a2 pointsLinear functionQuadratic functionCubic functionPolynomial of any degree
Solution
The Trapezoidal Rule gives the exact value of the integral when the integrand is a linear function. This is because the trapezoidal rule approximates the area under the curve by dividing it into trapezoids, and if the function is linear, the trapezoids perfectly fit under the curve. For higher degree functions like quadratic, cubic, or any other polynomial, the trapezoidal rule only provides an approximation.
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EXAMPLE 5 Use the Midpoint Rule with n = 5 to approximate the following integral.43x dx2SOLUTION The endpoints of the subintervals are 2, 2.4, 2.8, 3.2, 3.6, and 4, so the midpoints are 2.2, 2.6, 3, 3.4, and . The width of the subintervals is Δx = (4 − 2)/5 = , so the Midpoint Rule gives43x dx2≈ Δx[f(2.2) + f(2.6) + f(3) + f(3.4) + f(3.8)] = 0.432.2 + 32.6 + 33 + 33.4 + 33.8≈ . (Round your answer to four decimal places.)Since f(x) = 3x > 0 for 2 ≤ x ≤ 4, the integral represents an area, and the approximation given by the Midpoint Rule is the sum of the areas of the rectangles shown in the figure.
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