Use the trapezoidal rule to approximate the integral with $n$ trapezoids. Round your answer to the nearest thousandth.
$\int_1^6 \cos ^2 \frac{\pi}{x} dx \approx[?] \quad n=5$
Answer (1)
Calculate the width of each trapezoid: Δ x = 5 6 − 1 = 1 .
Evaluate the function f ( x ) = cos 2 ( x π ) at x i = 1 + i , for i = 0 , 1 , 2 , 3 , 4 , 5 .
Apply the trapezoidal rule formula: ∫ 1 6 cos 2 ( x π ) d x ≈ 2 Δ x [ f ( x 0 ) + 2 f ( x 1 ) + 2 f ( x 2 ) + 2 f ( x 3 ) + 2 f ( x 4 ) + f ( x 5 )] .
Approximate the integral and round to the nearest thousandth: 2.280 .
Explanation
Problem Setup We are asked to approximate the definite integral ∫ 1 6 cos 2 ( x π ) d x using the trapezoidal rule with n = 5 trapezoids. The trapezoidal rule provides an approximation of the definite integral by dividing the area under the curve into trapezoids and summing their areas.
Calculating Trapezoid Width First, we need to calculate the width of each trapezoid, which is given by the formula Δ x = n b − a , where a and b are the limits of integration and n is the number of trapezoids. In this case, a = 1 , b = 6 , and n = 5 . Therefore, Δ x = 5 6 − 1 = 5 5 = 1 .
Determining x-values Next, we need to determine the x values at which to evaluate the function f ( x ) = cos 2 ( x π ) . These values are given by x i = a + i Δ x , where i = 0 , 1 , 2 , 3 , 4 , 5 . So, we have: x 0 = 1 + 0 ( 1 ) = 1 x 1 = 1 + 1 ( 1 ) = 2 x 2 = 1 + 2 ( 1 ) = 3 x 3 = 1 + 3 ( 1 ) = 4 x 4 = 1 + 4 ( 1 ) = 5 $x_5 = 1 + 5(1) = 6
Evaluating the Function Now, we evaluate the function f ( x ) = cos 2 ( x π ) at each of these x i values: f ( x 0 ) = f ( 1 ) = cos 2 ( π ) = ( − 1 ) 2 = 1 f ( x 1 ) = f ( 2 ) = cos 2 ( 2 π ) = 0 2 = 0 f ( x 2 ) = f ( 3 ) = cos 2 ( 3 π ) = ( 2 1 ) 2 = 4 1 = 0.25 f ( x 3 ) = f ( 4 ) = cos 2 ( 4 π ) = ( 2 2 ) 2 = 2 1 = 0.5 f ( x 4 ) = f ( 5 ) = cos 2 ( 5 π ) ≈ 0.6545 f ( x 5 ) = f ( 6 ) = cos 2 ( 6 π ) = ( 2 3 ) 2 = 4 3 = 0.75
Applying the Trapezoidal Rule The trapezoidal rule formula is given by: ∫ a b f ( x ) d x ≈ 2 Δ x [ f ( x 0 ) + 2 f ( x 1 ) + 2 f ( x 2 ) + 2 f ( x 3 ) + 2 f ( x 4 ) + f ( x 5 )] Substituting the calculated values, we get: ∫ 1 6 cos 2 ( x π ) d x ≈ 2 1 [ 1 + 2 ( 0 ) + 2 ( 0.25 ) + 2 ( 0.5 ) + 2 ( 0.6545 ) + 0.75 ] ≈ 2 1 [ 1 + 0 + 0.5 + 1 + 1.309 + 0.75 ] ≈ 2 1 [ 4.559 ] ≈ 2.2795
Rounding the Result Finally, we round the result to the nearest thousandth: 2.2795 ≈ 2.280 .
Final Answer Therefore, the approximate value of the definite integral ∫ 1 6 cos 2 ( x π ) d x using the trapezoidal rule with n = 5 is 2.280 .
Examples
Imagine you are designing a solar panel and need to calculate the average solar energy absorption over a day. The solar energy absorption can be modeled by a function, and you can use the trapezoidal rule to approximate the integral of this function over the day's duration. This approximation helps you estimate the total energy absorbed by the solar panel, which is crucial for determining its efficiency and power output. By dividing the day into smaller intervals and applying the trapezoidal rule, you can get a reasonably accurate estimate of the total solar energy absorbed.
