Final answer:
SIMP(2) is an approximation for the integral ∫₀⁴ x³ dx using Simpson's Rule with two subintervals, where the final approximation of the integral is 64.
Step-by-step explanation:
The student is asking to find an approximation for the integral ∫₀⁴ x³ dx using the Simpson's Rule, denoted here as SIMP(2). Simpson's Rule is a method for approximating the value of a definite integral using a parabolic arc. To apply Simpson's Rule in this instance, the interval [0, 4] would be split into two equal subintervals (since SIMP(2) indicates two subintervals). The endpoints and the midpoint would be used. In this case, the endpoints are x=0 and x=4, and the midpoint is x=2.
First, we evaluate the function x³ at these points: f(0)=0³=0, f(2)=2³=8, and f(4)=4³=64. Then, applying Simpson's Rule: SIMP(2) = ∂x/3 [f(x₀) + 4f(x₁) + f(x₂)], where ∂x is the width of each subinterval (∂x = (b-a)/n; in this case: (4-0)/2 = 2), and x₀, x₁, and x₂ are the sample points used.
Therefore, the approximation SIMP(2) for the integral is:
SIMP(2) = (2/3) [f(0) + 4f(2) + f(4)]
= (2/3) [0 + 4(8) + 64]
= (2/3) [32 + 64]
= (2/3) [96]
= 64.
Thus, SIMP(2) for the integral ∫₀⁴ x³ dx is approximately equal to 64.