incorrect. the definite integral is over an interval of length 4, which is consistent with δx = 4/n, and the…

incorrect. the definite integral is over an interval of length 4, which is consistent with δx = 4/n, and the integral can be written over the interval 1, 5. however, if the lower limit is taken to be 1, then the endpoints that are being used for a right riemann sum are of the form xk = 1 + 4k/n = 1 + kδx with the last endpoint (k = n) at x = 5. the expression 12k/n, however, would be equal to 3(xk - 1), and the riemann sum would be of the form ∑k = 1n 3(xk - 1)cos(xk)δx = ∑k = 1n f(xk)δx, where f(x) = 3(x - 1)cos x. the limit of this sum would be written as ∫15 3(x - 1)cos x dx, not ∫15 3x cos x dx.

incorrect. the definite integral is over an interval of length 4, which is consistent with δx = 4/n, and the integral can be written over the interval 1, 5. however, if the lower limit is taken to be 1, then the endpoints that are being used for a right riemann sum are of the form xk = 1 + 4k/n = 1 + kδx with the last endpoint (k = n) at x = 5. the expression 12k/n, however, would be equal to 3(xk - 1), and the riemann sum would be of the form ∑k = 1n 3(xk - 1)cos(xk)δx = ∑k = 1n f(xk)δx, where f(x) = 3(x - 1)cos x. the limit of this sum would be written as ∫15 3(x - 1)cos x dx, not ∫15 3x cos x dx.

Answer

Explanation:

Step1: Analyze integral interval

The interval length is 4, $\Delta x=\frac{4}{n}$, and the interval is [1,5].

Step2: Determine right - Riemann sum endpoints

For a right - Riemann sum with lower limit 1, $x_k = 1 + k\Delta x=1+\frac{4k}{n}$, last endpoint at $x = 5$ when $k = n$.

Step3: Check the function in Riemann sum

The given expression $\frac{12k}{n}=3(x_k - 1)$. The Riemann sum is $\sum_{k = 1}^{n}3(x_k - 1)\cos(x_k)\Delta x=\sum_{k = 1}^{n}f(x_k)\Delta x$ with $f(x)=3(x - 1)\cos x$.

Step4: Find the corresponding definite integral

The limit of the sum is $\int_{1}^{5}3(x - 1)\cos xdx$, not $\int_{1}^{5}3x\cos xdx$.

Answer:

The analysis shows the correct definite - integral form for the given right - Riemann sum setup should be $\int_{1}^{5}3(x - 1)\cos xdx$ instead of $\int_{1}^{5}3x\cos xdx$.