Approximate the integral by the trapezoidal rule.
-$\int _ { 4 } ^ { 6 } ( \ln x ) ^ { 6 }$ dx; n = 4 Express your answer to five decimal places.
A) 23.42223
B) 147.19132
C) 36.79783
D) 73.59566

Question

Approximate the integral by the trapezoidal rule. 
-$\int _ { 4 } ^ { 6 } ( \ln x ) ^ { 6 }$ dx; n = 4 Express your answer to five decimal places.
A) 23.42223
B) 147.19132
C) 36.79783
D) 73.59566

Quizplus · Accepted Answer

The trapezoidal rule formula for approximating an integral $\int_a^b f(x) dx$ with $n$ subintervals is given by $\frac{\Delta x}{2} [f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n)]$, where $\Delta x = \frac{b-a}{n}$ and $x_i = a + i\Delta x$. For $\int _ { 4 } ^ { 6 } ( \ln x ) ^ { 6 }$ dx with $n = 4$, we have $a = 4$, $b = 6$, and $\Delta x = \frac{6-4}{4} = 0.5$. Calculating the values of the function at the relevant points and applying the formula, we find that the approximation is closest to 36.79783.

Approximate the Integral by the Trapezoidal Rule ∫46(ln⁡x)6\int _ { 4 } ^ { 6 } ( \ln x ) ^ { 6 }∫46​(lnx)6

Approximate the Integral by the Trapezoidal Rule $\int _ { 4 } ^ { 6 } ( \ln x ) ^ { 6 }$