Calculate the area under the curve of f(x) = 2x2+ 1, within the interval [0, 2], using the method of trapezoids. The number of trapezoids must be 4.

To calculate the area under the curve of the function f(x) = 2x^2 + 1 within the interval [0, 2] using the method of trapezoids, we can divide the interval into four equal subintervals and approximate the area using the trapezoidal rule. The trapezoidal rule formula for n = 4 subintervals is given as: A ≈ (Δx/2)[f(x0) + 2f(x1) + 2f(x2) + 2f(x3) + f(x4)] Here, the subinterval width Δx = (b - a)/n = (2 - 0)/4 = 0.5, where a = 0 and b = 2. Substituting the values from the function into the formula, we have: A ≈ (0.5/2)[f(0) + 2f(0.5) + 2f(1) + 2f(1.5) + f(2)] A ≈ (0.5/2)[1 + 2(2.5) + 2(5) + 2(9) + 17] Simplifying the expression, we find: A ≈ (0.5/2)[1 + 5 + 10 + 18 + 17] A ≈ (0.5/2)[51] A ≈ 12.75 Therefore, the area under the curve of the function f(x) = 2x^2 + 1 within the interval [0, 2] using the method of trapezoids with four trapezoids is approximately 12.75.