Use Simpson's Rule with 10 subintervals to estimate the area under the graph of y = 0.2x2 for −4 ≤ x ≤ 2.

Sure, let's use Simpson's Rule to estimate the area under the graph of y = 0.2x^2 for -4 ≤ x ≤ 2.

Simpson's Rule is a numerical method for approximating the definite integral of a function. It's given by the formula:

∫(from a to b) f(x) dx ≈ (b - a) / 6n [f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + 2f(x4) + ... + 4f(xn-1) + f(xn)]

where n is the number of subintervals, and x0, x1, ..., xn are the endpoints of the subintervals.

Here, a = -4, b = 2, and n = 10. So, the width of each subinterval is h = (b - a) / n = (2 - (-4)) / 10 = 0.6.

Now, we need to calculate the values of the function f at the endpoints of the subintervals:

f(x0) = f(-4) = 0.2 * (-4)^2 = 3.2, f(x1) = f(-3.4) = 0.2 * (-3.4)^2 = 2.312, f(x2) = f(-2.8) = 0.2 * (-2.8)^2 = 1.568, ... f(x10) = f(2) = 0.2 * 2^2 = 0.8.

Finally, we substitute these values into Simpson's Rule formula:

∫(from -4 to 2) 0.2x^2 dx ≈ 0.6 / 6 * [3.2 + 42.312 + 21.568 + ... + 40.8 + 0.8] = 0.1 * [3.2 + 42.312 + 21.568 + ... + 40.8 + 0.8].

You can calculate the sum inside the brackets and multiply it by 0.1 to get the final result.