approximate the value of the integral by use of the trapezoidal rule. n=8

a=0
b=4
sqrt 16-x^2

To approximate the value of the integral using the trapezoidal rule, follow these steps:

1. Divide the interval [a, b] into n equal subintervals. In this case, a = 0, b = 4, and n = 8. So each subinterval has a width of (b - a) / n = (4 - 0) / 8 = 0.5.

2. Evaluate the function at the endpoints of each subinterval. In this case, we need to evaluate the function f(x) = √(16 - x^2) at the endpoints of each subinterval. Since there are 8 subintervals, we need to evaluate at 9 points: x0 = 0, x1 = 0.5, x2 = 1.0, ..., x8 = 4.0.

3. Calculate the sum of the function evaluations at these points. The formula to calculate the approximation using the trapezoidal rule is:
Approximation ≈ (h / 2) * [f(x0) + 2f(x1) + 2f(x2) + ... + 2f(xn-1) + f(xn)]
where h is the width of each subinterval (0.5 in this case), and f(x0), f(x1), ..., f(xn) are the function values at the specified points.

4. Substitute the function values into the formula and calculate the approximation.

Approximation ≈ (0.5 / 2) * [f(0) + 2f(0.5) + 2f(1) + 2f(1.5) + 2f(2) + 2f(2.5) + 2f(3) + 2f(3.5) + f(4)]

To find the function values at each point, substitute the value of x into the function f(x) = √(16 - x^2).

f(0) = √(16 - 0^2) = 4
f(0.5) = √(16 - 0.5^2) ≈ 3.998
f(1) = √(16 - 1^2) ≈ 3.874
f(1.5) = √(16 - 1.5^2) ≈ 3.671
f(2) = √(16 - 2^2) ≈ 3.464
f(2.5) = √(16 - 2.5^2) ≈ 3.235
f(3) = √(16 - 3^2) ≈ 2.989
f(3.5) = √(16 - 3.5^2) ≈ 2.654
f(4) = √(16 - 4^2) ≈ 2

Now, substitute these values into the formula and calculate the approximation:

Approximation ≈ (0.5 / 2) * [4 + 2(3.998) + 2(3.874) + 2(3.671) + 2(3.464) + 2(3.235) + 2(2.989) + 2(2.654) + 2]

Approximation ≈ 6.0

Therefore, the approximate value of the integral using the trapezoidal rule with n = 8 is 6.0.