Final answer:
To estimate the integral ∫201x³ √2 dx by the trapezoidal rule using n = 4, divide the interval [0, 2] into 4 subintervals, calculate the area of each trapezoid, and sum them up.
Step-by-step explanation:
The trapezoidal rule is a numerical method used to estimate the value of a definite integral. To estimate the integral ∫201x³ √2 dx by the trapezoidal rule with n = 4, we divide the interval [0, 2] into 4 equal subintervals: [0, 0.5], [0.5, 1], [1, 1.5], [1.5, 2]. We then calculate the area of each trapezoid and sum them up to estimate the integral.
- Calculate the width of each subinterval: Δx = (b - a) / n = (2 - 0) / 4 = 0.5
- Calculate the values of the function at each point: f(0) = 201(0)³ √2 = 0, f(0.5) = 201(0.5)³ √2, f(1) = 201(1)³ √2, f(1.5) = 201(1.5)³ √2, f(2) = 201(2)³ √2 = 402√2
- Calculate the area of each trapezoid: Area1 = (f(0) + f(0.5)) * Δx / 2, Area2 = (f(0.5) + f(1)) * Δx / 2, Area3 = (f(1) + f(1.5)) * Δx / 2, Area4 = (f(1.5) + f(2)) * Δx / 2
- Sum up the areas of all the trapezoids: Estimate = Area1 + Area2 + Area3 + Area4
Plug in the respective values and follow the steps to find the estimate of the integral ∫201x³ √2 dx.