Use the Trapezoidal Rule with n=3 to approximate the value of ∫822x3dx .
To use the trapezoidal rule with n=3, we first need to divide the interval [8,22] into 3 subintervals of equal width. This will give us the subintervals [8, 14], [14, 18], and [18, 22].
Next, we compute the height of the trapezoid for each subinterval by evaluating the function at the endpoints of the subinterval. Then, we use the trapezoidal rule formula to calculate the approximate value of the integral:
∫822x^3dx ≈ (width/2) * [f(a) + 2f(a1) + 2f(a2) + f(b)]
where:
- width = (22-8)/3 = 14/3
- a = 8
- a1 = 14
- a2 = 18
- b = 22
- f(x) = x^3
Substitute these values into the formula and calculate:
≈ (14/6) * [f(8) + 2f(14) + 2f(18) + f(22)]
≈ (14/6) * [8^3 + 2(14^3) + 2(18^3) + 22^3]
≈ (14/6) * [512 + 2(2744) + 2(5832) + 10648]
≈ (14/6) * [512 + 5488 + 11664 + 10648]
≈ (14/6) * 27312
≈ 63 * 14
≈ 882
Therefore, the approximate value of ∫822x^3dx using the trapezoidal rule with n=3 is 882.