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Example
Created on 2024-06-20Asked by Theodore Green (Solvelet student)
Approximate the integral ∫01x2 dx \int_0^1 x^2 \, dx using the trapezoidal rule with n=4 n = 4 subintervals.

Solution

To approximate the integral ∫01x2 dx \int_0^1 x^2 \, dx using the trapezoidal rule with n=4 n = 4 subintervals, follow these steps: 1. Determine the width of each subinterval: h=b−an=1−04=0.25. h = \frac{b - a}{n} = \frac{1 - 0}{4} = 0.25. 2. Calculate the function values at the endpoints and midpoints: f(0)=02=0, f(0) = 0^2 = 0, f(0.25)=(0.25)2=0.0625, f(0.25) = (0.25)^2 = 0.0625, f(0.5)=(0.5)2=0.25, f(0.5) = (0.5)^2 = 0.25, f(0.75)=(0.75)2=0.5625, f(0.75) = (0.75)^2 = 0.5625, f(1)=12=1. f(1) = 1^2 = 1. 3. Apply the trapezoidal rule: ∫01x2 dx≈h2[f(0)+2f(0.25)+2f(0.5)+2f(0.75)+f(1)]. \int_0^1 x^2 \, dx \approx \frac{h}{2} [f(0) + 2f(0.25) + 2f(0.5) + 2f(0.75) + f(1)]. =0.252[0+2(0.0625)+2(0.25)+2(0.5625)+1], = \frac{0.25}{2} [0 + 2(0.0625) + 2(0.25) + 2(0.5625) + 1], =0.125[0+0.125+0.5+1.125+1], = 0.125 [0 + 0.125 + 0.5 + 1.125 + 1], =0.125×2.75=0.34375. = 0.125 \times 2.75 = 0.34375. Therefore, the approximate value of the integral using the trapezoidal rule is 0.34375 0.34375 . Solved on Solvelet with Basic AI Model
Some of the related questions asked by Mason Adams on Solvelet
1. Approximate the integral of the function f(x)=sin⁡(x) f(x) = \sin(x) over the interval [0,π] [0, \pi] using the trapezoidal rule with 4 subintervals.2. Estimate the area under the curve y=x2y = x^2 from x=0x = 0 to x=1x = 1 using Simpson's rule with 66 subintervals.
DefinitionFor the purpose of numerical integration, values of definite integrals are calculated based on discrete data points or values. Such techniques are the trapezoidal rule, Simpson's rule, and numerical quadrature. For instance, if f(x)=x2 and we wish to approximate ∫01​x2dx using the trapezoidal rule with n=2 intervals, we have∫01​x2dx≈21​[(02+12)+2⋅(21​)2]=21​(0+1+2⋅0.25)=0.375.
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