Use the Trapezoidal Rule with n=3 to approximate the value of ∫822x3dx . (1 point) Responses 1,152 1,152 1,980 1,980 2,070 2,070 2,160 2,160 3,168
To use the Trapezoidal Rule with n=3 to approximate the value of ∫822x3dx, first divide the interval [8, 22] into 3 subintervals of equal width.
The width of each subinterval, Δx, is calculated as (b-a)/n = (22-8)/3 = 14/3 = 4.6667.
Next, calculate the approximations of the integral using the Trapezoidal Rule:
∫822x^3dx ≈ Δx/2 * [f(a) + 2∑f(xi) + f(b)]
where f(x) = x^3.
Substituting the values into the formula:
≈ 4.6667/2 * [f(8) + 2(f(12) + f(16)) + f(22)]
≈ 4.6667/2 * [512 + 2(1728 + 4096) + 10648]
≈ 2.33335 * [512 + 2(1728 + 4096) + 10648]
≈ 2.33335 * [512 + 2(5824) + 10648]
≈ 2.33335 * [512 + 11648 + 10648]
≈ 2.33335 * 22808
≈ 53150.8888
Therefore, the approximate value of ∫822x^3dx using the Trapezoidal Rule with n=3 is 53,150. MainActivity8888.