Final answer:
The integral ∫₀¹ (1/1+x²) dx is equal to π/4, since it corresponds to the inverse tangent function. Using the trapezoidal rule with n=3 subdivisions gives an approximation of the integral, which is then used to estimate the value of π.
Step-by-step explanation:
To evaluate the integral ∫₀¹ (1/1+x²) dx and show that it is π/4, we recognize the integrand as the derivative of the inverse tangent function, arctan(x). Thus, the integral can be solved using a direct antiderivative:
∫₀¹ (1/1+x²) dx = arctan(1) - arctan(0) = π/4 - 0 = π/4.
To approximate the same integral using the trapezoidal rule with n=3 subdivisions, one would calculate the values of the function at the points x=0, x=1/3, x=2/3, and x=1, then apply the trapezoidal rule formula:
Approximate Integral = (1/2n) [f(0) + 2(f(1/3) + f(2/3)) + f(1)]
Plugging in the values, we get:
Approximate Integral = (½)(1/3) [(1) + 2(1/(1+(1/3)²) + 1/(1+(2/3)²)) + 1/(1+1²)] = 0.7853981...
Finally, this approximate integral can be used to estimate the value of π by multiplying the result by 4, since we have shown that the exact integral is π/4.
An approximate value of π is thus 4 * 0.7853981 = 3.1415924.