Problem 77 - Calculus Early Transcendentals

Question

Use numerical methods or a calculator to approximate the following integrals as closely as possible. The exact value of each integral is given. $$\int_{0}^{\pi / 2} \ln (\sin x) d x=\int_{0}^{\pi / 2} \ln (\cos x) d x=-\frac{\pi \ln 2}{2}$$

Step-by-Step Solution

Answer

**Question:** Using the trapezoidal rule, approximate the value of the following integrals: 1. $$\int_{0}^{\pi / 2} \ln (\sin x) d x$$ 2. $$\int_{0}^{\pi / 2} \ln (\cos x) d x$$ Compare the approximation with the exact value, $$-\frac{\pi \ln 2}{2}$$.

Step 1: Divide the interval

Divide the interval [0, π/2] into n equally spaced subintervals. The width of each subinterval is $$\Delta x = \frac{\pi / 2 - 0}{n}$$.

Step 2: Calculate the function values

Calculate the function value, $$\ln(\sin x)$$, at the endpoints of each subinterval.

Step 3: Apply the trapezoidal rule

Use the trapezoidal rule to approximate the integral: $$\int_{0}^{\pi / 2} \ln (\sin x) d x \approx \frac{\Delta x}{2}\left[ \ln (\sin 0) + 2 \sum_{i=1}^{n-1} \ln (\sin i \Delta x) + \ln (\sin \frac{\pi}{2}) \right]$$ Now, let's approximate the second integral using the trapezoidal rule.

Step 4: Divide the interval

Divide the interval [0, π/2] into n equally spaced subintervals. The width of each subinterval is $$\Delta x = \frac{\pi / 2 - 0}{n}$$.

Step 5: Calculate the function values

Calculate the function value, $$\ln(\cos x)$$, at the endpoints of each subinterval.

Step 6: Apply the trapezoidal rule

Use the trapezoidal rule to approximate the integral: $$\int_{0}^{\pi / 2} \ln (\cos x) d x \approx \frac{\Delta x}{2}\left[ \ln (\cos 0) + 2 \sum_{i=1}^{n-1} \ln (\cos i \Delta x) + \ln (\cos \frac{\pi}{2}) \right]$$ Finally, compare the approximated values of the integrals with the given exact value, $$-\frac{\pi \ln 2}{2}$$.