Find the Riemann sum for \(\displaystyle f(x) = 4 \sin x\), over interval \(\displaystyle [0, \dfrac{3\pi}{2}]\) with six terms, taking the sample points to be midpoints. (Round your answers to six decimal places.) Find \(\displaystyle M_{6}\) - the sixth midpoint.
\(\displaystyle \sum\limits_{i=6}^n \Delta x [f(a + i \Delta x) - \dfrac{1}{2} \Delta x]\)
\(\displaystyle \Delta x = \dfrac{b - a}{n}\)
\(\displaystyle \Delta x = \dfrac{\dfrac{3\pi}{2} - 0}{6} = \dfrac{\pi}{4}\)
\(\displaystyle n = 6\)
\(\displaystyle \sum\limits_{i=6}^n (\dfrac{\pi}{4}) [[f(0 +(1)(\dfrac{\pi}{4}) - \dfrac{1}{2} (\dfrac{\pi}{4}))] + [f(0 +(2)(\dfrac{\pi}{4}) - \dfrac{1}{2} (\dfrac{\pi}{4}))] + [f(0 +(3)(\dfrac{\pi}{4}) - \dfrac{1}{2} (\dfrac{\pi}{4}))] + [f(0 +(4)(\dfrac{\pi}{4}) - \dfrac{1}{2} (\dfrac{\pi}{4})) ] + [f(0 + (5)(\dfrac{\pi}{4}) - \dfrac{1}{2} (\dfrac{\pi}{4}))] + [f(0 + ( 6)(\dfrac{\pi}{4}) - \dfrac{1}{2} (\dfrac{\pi}{4})) ]]\)
\(\displaystyle \sum\limits_{i=6}^n (\dfrac{\pi}{4}) [[f(0 +(1)(\dfrac{\pi}{4}) - \dfrac{\pi}{8})] + [f(0 +(2)(\dfrac{\pi}{4}) - \dfrac{\pi}{8} )] + [f(0 +(3)(\dfrac{\pi}{4}) - \dfrac{\pi}{8})] + [f(0 +(4)(\dfrac{\pi}{4}) - \dfrac{\pi}{8}) ] + [f(0 + (5)(\dfrac{\pi}{4}) - \dfrac{\pi}{8})] + [f(0 + ( 6)(\dfrac{\pi}{4}) - \dfrac{\pi}{8}) ]]\)
\(\displaystyle \sum\limits_{i=6}^n (\dfrac{\pi}{4}) [[f(0 + (\dfrac{\pi}{4}) - \dfrac{\pi}{8})] + [f(0 + (\dfrac{2\pi}{4}) - \dfrac{\pi}{8} )] + [f(0 + (\dfrac{3\pi}{4}) - \dfrac{\pi}{8})] + [f(0 + (\dfrac{4\pi}{4}) - \dfrac{\pi}{8}) ] + [f(0 + (\dfrac{5\pi}{4}) - \dfrac{\pi}{8})] + [f(0 + (\dfrac{6\pi}{4}) - \dfrac{\pi}{8}) ]]\)
\(\displaystyle \sum\limits_{i=6}^n (\dfrac{\pi}{4}) [[f(0 + (\dfrac{\pi}{4}) - \dfrac{\pi}{8})] + [f(0 + (\dfrac{\pi}{2}) - \dfrac{\pi}{8} )] + [f(0 + (\dfrac{3\pi}{4}) - \dfrac{\pi}{8})] + [f(0 + (\pi) - \dfrac{\pi}{8}) ] + [f(0 + (\dfrac{5\pi}{4}) - \dfrac{\pi}{8})] + [f(0 + (\dfrac{3\pi}{2}) - \dfrac{\pi}{8}) ]]\)
\(\displaystyle \sum\limits_{i=6}^n (\dfrac{\pi}{4}) [[f(0 + \dfrac{\pi}{8})] + [f(0 + \dfrac{3\pi}{8} )] + [f(0 + \dfrac{5\pi}{8})] + [f(0 + \dfrac{7\pi}{8}) ] + [f(0 + \dfrac{9\pi}{8})] + [f(0 + \dfrac{11\pi}{8}) ]]\)
\(\displaystyle \sum\limits_{i=6}^n (\dfrac{\pi}{4}) [[f(\dfrac{\pi}{8})] + [f(\dfrac{3\pi}{8} )] + [f(\dfrac{5\pi}{8})] + [f(\dfrac{7\pi}{8}) ] + [f(\dfrac{9\pi}{8})] + [f(\dfrac{11\pi}{8}) ]]\)
\(\displaystyle \sum\limits_{i=6}^n (\dfrac{\pi}{4}) [[4 \sin (\dfrac{\pi}{8})] + [4 \sin (\dfrac{3\pi}{8} )] + [4 \sin (\dfrac{5\pi}{8})] + [4 \sin (\dfrac{7\pi}{8}) ] + [4 \sin (\dfrac{9\pi}{8})] + [4 \sin (\dfrac{11\pi}{8}) ]]\)
\(\displaystyle \sum\limits_{i=6}^n (\dfrac{\pi}{4}) [[4 (.3827)] + [4 (.9239 )] + [4 (.9239)] + [4 (.3827) ] + [4 (-.3827)] + [4 (-.9239) ]]\)
\(\displaystyle \sum\limits_{i=6}^n (\dfrac{\pi}{4}) [[1.5308] + [3.6956] + [3.6956] + [1.5308 ] + [-1.5308] + [-3.6956 ]]\)
\(\displaystyle \sum\limits_{i=6}^n (.7854) [[1.5308] + [3.6956] + [3.6956] + [1.5308 ] + [-1.5308] + [-3.6956 ]]\)
\(\displaystyle \sum\limits_{i=6}^n [1.2017] + [2.9025] + [2.9025] + [1.2017 ] + [-1.2017] + [-2.9025 ]\) - on the right track?
\(\displaystyle \sum\limits_{i=6}^n \Delta x [f(a + i \Delta x) - \dfrac{1}{2} \Delta x]\)
\(\displaystyle \Delta x = \dfrac{b - a}{n}\)
\(\displaystyle \Delta x = \dfrac{\dfrac{3\pi}{2} - 0}{6} = \dfrac{\pi}{4}\)
\(\displaystyle n = 6\)
\(\displaystyle \sum\limits_{i=6}^n (\dfrac{\pi}{4}) [[f(0 +(1)(\dfrac{\pi}{4}) - \dfrac{1}{2} (\dfrac{\pi}{4}))] + [f(0 +(2)(\dfrac{\pi}{4}) - \dfrac{1}{2} (\dfrac{\pi}{4}))] + [f(0 +(3)(\dfrac{\pi}{4}) - \dfrac{1}{2} (\dfrac{\pi}{4}))] + [f(0 +(4)(\dfrac{\pi}{4}) - \dfrac{1}{2} (\dfrac{\pi}{4})) ] + [f(0 + (5)(\dfrac{\pi}{4}) - \dfrac{1}{2} (\dfrac{\pi}{4}))] + [f(0 + ( 6)(\dfrac{\pi}{4}) - \dfrac{1}{2} (\dfrac{\pi}{4})) ]]\)
\(\displaystyle \sum\limits_{i=6}^n (\dfrac{\pi}{4}) [[f(0 +(1)(\dfrac{\pi}{4}) - \dfrac{\pi}{8})] + [f(0 +(2)(\dfrac{\pi}{4}) - \dfrac{\pi}{8} )] + [f(0 +(3)(\dfrac{\pi}{4}) - \dfrac{\pi}{8})] + [f(0 +(4)(\dfrac{\pi}{4}) - \dfrac{\pi}{8}) ] + [f(0 + (5)(\dfrac{\pi}{4}) - \dfrac{\pi}{8})] + [f(0 + ( 6)(\dfrac{\pi}{4}) - \dfrac{\pi}{8}) ]]\)
\(\displaystyle \sum\limits_{i=6}^n (\dfrac{\pi}{4}) [[f(0 + (\dfrac{\pi}{4}) - \dfrac{\pi}{8})] + [f(0 + (\dfrac{2\pi}{4}) - \dfrac{\pi}{8} )] + [f(0 + (\dfrac{3\pi}{4}) - \dfrac{\pi}{8})] + [f(0 + (\dfrac{4\pi}{4}) - \dfrac{\pi}{8}) ] + [f(0 + (\dfrac{5\pi}{4}) - \dfrac{\pi}{8})] + [f(0 + (\dfrac{6\pi}{4}) - \dfrac{\pi}{8}) ]]\)
\(\displaystyle \sum\limits_{i=6}^n (\dfrac{\pi}{4}) [[f(0 + (\dfrac{\pi}{4}) - \dfrac{\pi}{8})] + [f(0 + (\dfrac{\pi}{2}) - \dfrac{\pi}{8} )] + [f(0 + (\dfrac{3\pi}{4}) - \dfrac{\pi}{8})] + [f(0 + (\pi) - \dfrac{\pi}{8}) ] + [f(0 + (\dfrac{5\pi}{4}) - \dfrac{\pi}{8})] + [f(0 + (\dfrac{3\pi}{2}) - \dfrac{\pi}{8}) ]]\)
\(\displaystyle \sum\limits_{i=6}^n (\dfrac{\pi}{4}) [[f(0 + \dfrac{\pi}{8})] + [f(0 + \dfrac{3\pi}{8} )] + [f(0 + \dfrac{5\pi}{8})] + [f(0 + \dfrac{7\pi}{8}) ] + [f(0 + \dfrac{9\pi}{8})] + [f(0 + \dfrac{11\pi}{8}) ]]\)
\(\displaystyle \sum\limits_{i=6}^n (\dfrac{\pi}{4}) [[f(\dfrac{\pi}{8})] + [f(\dfrac{3\pi}{8} )] + [f(\dfrac{5\pi}{8})] + [f(\dfrac{7\pi}{8}) ] + [f(\dfrac{9\pi}{8})] + [f(\dfrac{11\pi}{8}) ]]\)
\(\displaystyle \sum\limits_{i=6}^n (\dfrac{\pi}{4}) [[4 \sin (\dfrac{\pi}{8})] + [4 \sin (\dfrac{3\pi}{8} )] + [4 \sin (\dfrac{5\pi}{8})] + [4 \sin (\dfrac{7\pi}{8}) ] + [4 \sin (\dfrac{9\pi}{8})] + [4 \sin (\dfrac{11\pi}{8}) ]]\)
\(\displaystyle \sum\limits_{i=6}^n (\dfrac{\pi}{4}) [[4 (.3827)] + [4 (.9239 )] + [4 (.9239)] + [4 (.3827) ] + [4 (-.3827)] + [4 (-.9239) ]]\)
\(\displaystyle \sum\limits_{i=6}^n (\dfrac{\pi}{4}) [[1.5308] + [3.6956] + [3.6956] + [1.5308 ] + [-1.5308] + [-3.6956 ]]\)
\(\displaystyle \sum\limits_{i=6}^n (.7854) [[1.5308] + [3.6956] + [3.6956] + [1.5308 ] + [-1.5308] + [-3.6956 ]]\)
\(\displaystyle \sum\limits_{i=6}^n [1.2017] + [2.9025] + [2.9025] + [1.2017 ] + [-1.2017] + [-2.9025 ]\) - on the right track?
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