Approximate the definite integral by right Riemann sum with the indicated partitions. {:[y=cos x","],[[0","pi//6","pi//4","pi//2","3pi//4","pi]]:}

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Approximate the definite integral by right Riemann sum with the indicated partitions.  {:[y=cos x","],[[0","pi//6","pi//4","pi//2","3pi//4","pi]]:}

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Problem and Riemann sum setup: Understand the problem and set up the Riemann sum. We are asked to approximate the definite integral of the function y = cos(x) over the interval [0, pi] using a right Riemann sum. This means we will use the value of the function at the right endpoint of each subinterval to calculate the sum. The partitions given are [0, pi/6, pi/4, pi/2, 3pi/4, pi], which divide the interval [0, pi] into 5 subintervals. We will calculate the width of each subinterval and the value of the function at the right endpoint of each subinterval.Calculate subinterval widths: Calculate the width of each subinterval. The widths of the subintervals are as follows: - From pi/6 to pi/4: pi/4 - pi/6 = pi/12 - From pi/4 to pi/2: pi/2 - pi/4 = pi/4 - From pi/2 to 3pi/4: 3pi/4 - pi/2 = pi/4 - From 3pi/4 to pi: pi - 3pi/4 = pi/4 Note that the first subinterval (from 0 to pi/6) is not used in the right Riemann sum because we are using the right endpoints, and 0 is the left endpoint of the first subinterval.Evaluate function at right endpoints: Evaluate the function at the right endpoints of each subinterval. We will evaluate cos(x) at the right endpoints of each subinterval: - cos(pi/4) = sqrt(2)/2 - cos(pi/2) = 0 - cos(3pi/4) = -sqrt(2)/2 - cos(pi) = -1Multiply function values by subinterval widths: Multiply each function value by the width of its subinterval. Now we multiply the value of the function at each right endpoint by the width of the corresponding subinterval: - cos(pi/4) * (pi/4 - pi/6) = (sqrt(2)/2) * (pi/12) - cos(pi/2) * (pi/2 - pi/4) = 0 * (pi/4) - cos(3pi/4) * (3pi/4 - pi/2) = (-sqrt(2)/2) * (pi/4) - cos(pi) * (pi - 3pi/4) = -1 * (pi/4)Calculate right Riemann sum: Add up the products to get the right Riemann sum. The right Riemann sum is the sum of all the products calculated in the previous step: R = (sqrt(2)/2) * (pi/12) + 0 * (pi/4) + (-sqrt(2)/2) * (pi/4) + (-1) * (pi/4) R = (sqrt(2)/2) * (pi/12) - (sqrt(2)/2) * (pi/4) - (pi/4)Simplify Riemann sum: Simplify the Riemann sum. To simplify the Riemann sum, we combine like terms and perform the arithmetic: R = (sqrt(2)/2) * (pi/12) - (sqrt(2)/2) * (3pi/12) - (pi/4) R = (sqrt(2)/2) * (pi/12 - 3pi/12) - (pi/4) R = (sqrt(2)/2) * (-2pi/12) - (pi/4) R = (-sqrt(2)/6) * (pi) - (pi/4) R = -sqrt(2)pi/6 - pi/4

Approximate the definite integral by right Riemann sum with the indicated partitions. $\newline$ $y=\cos x$ $\newline$ $[0,\frac{\pi}{6},\frac{\pi}{4},\frac{\pi}{2},\frac{3\pi}{4},\pi]$

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Approximate the definite integral by right Riemann sum with the indicated partitions. $\newline$ $y=\cos x$ $\newline$ $[0,\frac{\pi}{6},\frac{\pi}{4},\frac{\pi}{2},\frac{3\pi}{4},\pi]$