Use $n=4$ to approximate the value of the integral by the following methods: (a) the trapezoidal rule, and (b) Simpson's rule. (c) Find the exact value by integration. \[ \int_{1}^{4} \frac{9}{x^{2}} d x \] \[ \int_{1} \frac{9}{x^{2}} d x \approx \] (Round the final answer to three decimal places as needed. Round all intermediate values to four decimal places as needed) (b) Use Simpson's rule to approximate the integral. \[ \int_{1}^{4} \frac{9}{x^{2}} d x \approx \] (Round the final answer to three decimal places as needed. Round all intermediate values to four decimal places as needed) (c) Find the exact value of the integral by integration \[ \int_{1}^{4} \frac{9}{x^{2}} d x= \] (Type an integer or a decimal Do not round.)

Step 1 ：First, we need to calculate the width of each subinterval. The width \(h\) is given by \((b - a) / n\), where \(a\) and \(b\) are the limits of integration and \(n\) is the number of subintervals. In this case, \(a = 1\), \(b = 4\), and \(n = 4\), so \(h = (4 - 1) / 4 = 0.75\).

Step 2 ：(a) To approximate the integral using the trapezoidal rule, we use the formula \(\int_{a}^{b} f(x) dx \approx h/2 [f(x_0) + 2f(x_1) + 2f(x_2) + ... + 2f(x_{n-1}) + f(x_n)]\), where \(x_i = a + ih\). Here, \(f(x) = 9/x^2\), so we calculate \(f(x_i)\) for \(i = 0, 1, 2, 3, 4\).

Step 3 ：We find that \(f(x_0) = 9/1^2 = 9\), \(f(x_1) = 9/1.75^2 = 2.9388\), \(f(x_2) = 9/2.5^2 = 1.44\), \(f(x_3) = 9/3.25^2 = 0.8527\), and \(f(x_4) = 9/4^2 = 0.5625\).

Step 4 ：Substituting these values into the trapezoidal rule formula, we get \(\int_{1}^{4} 9/x^2 dx \approx 0.75/2 [9 + 2(2.9388) + 2(1.44) + 2(0.8527) + 0.5625] = 6.75\).

Step 5 ：(b) To approximate the integral using Simpson's rule, we use the formula \(\int_{a}^{b} f(x) dx \approx h/3 [f(x_0) + 4f(x_1) + 2f(x_2) + ... + 4f(x_{n-1}) + f(x_n)]\), where \(x_i = a + ih\). Here, \(f(x) = 9/x^2\), so we calculate \(f(x_i)\) for \(i = 0, 1, 2, 3, 4\).

Step 6 ：We find that \(f(x_0) = 9/1^2 = 9\), \(f(x_1) = 9/1.75^2 = 2.9388\), \(f(x_2) = 9/2.5^2 = 1.44\), \(f(x_3) = 9/3.25^2 = 0.8527\), and \(f(x_4) = 9/4^2 = 0.5625\).

Step 7 ：Substituting these values into the Simpson's rule formula, we get \(\int_{1}^{4} 9/x^2 dx \approx 0.75/3 [9 + 4(2.9388) + 2(1.44) + 4(0.8527) + 0.5625] = 6.75\).

Step 8 ：(c) To find the exact value of the integral, we use the antiderivative of \(f(x) = 9/x^2\), which is \(-9/x\). The definite integral from \(a\) to \(b\) is given by \(\int_{a}^{b} f(x) dx = F(b) - F(a)\), where \(F(x)\) is the antiderivative of \(f(x)\).

Step 9 ：We find that \(F(4) = -9/4 = -2.25\) and \(F(1) = -9/1 = -9\). So, \(\int_{1}^{4} 9/x^2 dx = F(4) - F(1) = -2.25 - (-9) = 6.75\).

Step 10 ：So, the approximate value of the integral using the trapezoidal rule is \(\boxed{6.75}\), the approximate value using Simpson's rule is \(\boxed{6.75}\), and the exact value is \(\boxed{6.75}\).