Question

Use the Midpoint Rule with \(n=4\) to approximate the area of the region. Compare your result with the exact area obtained with a definite integral. $$ f(x)=1-x^{2}, \quad[-1,1] $$

Short answer

The estimated area under the curve \(f(x) = 1-x^2\) over the interval [-1,1] using the Midpoint Rule is 1.5. The exact area under the curve is 1.333.

Step-by-step solution

Use the Midpoint Rule

The Midpoint Rule states that the integral of a function on an interval [a,b] can be approximated by \((b - a) \cdot f((a + b) / 2)\), divided equally amongst n intervals. In this case, a = -1, b = 1, n = 4. So the length of each subinterval, \(\Delta x = (b - a) / n = (1 - (-1)) / 4 = 0.5\). The midpoints of each subinterval are -0.75, -0.25, 0.25, 0.75. Thus, the estimate using the Midpoint Rule is: \(0.5 [f(-0.75) + f(-0.25) + f(0.25) + f(0.75)]\). Substituting the function \(f(x) = 1-x^2\) into the formula gives the result: \(0.5 [(1-(-0.75)^2) + (1-(-0.25)^2) + (1-0.25^2) + (1-0.75^2)] = 0.5 [1 - 0.5625 + 1 - 0.0625 + 1 - 0.0625 + 1 - 0.5625] = 1.5.\)

Calculate the exact area under the curve

The exact area under the curve \(f(x) = 1-x^{2}\) over the interval [-1,1] can be found using a definite integral: \(\int_{-1}^1 (1-x^{2}) dx\). The antiderivative of \(1-x^{2}\) is \(x - x^3/3\). By the fundamental theorem of calculus, the exact area is \(F(b) - F(a)\), where \(F(x)\) is the antiderivative. Substituting the terminals into the integral: \([1 - 1^3/3] - [-1 - (-1)^3/3] = 2 - 2/3 = 1.333.\)

Compare the results

From step 1, the estimated area under the curve using the Midpoint Rule is 1.5. From step 2, the exact area under the curve found with a definite integral is 1.333. There is a small difference between the Midpoint Rule approximation and the exact value, as expected with numerical approximations.