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 right endpoints. (Round your answers to six decimal places.) Find \(\displaystyle R_{6}\) - the sixth right endpoint.
\(\displaystyle \sum\limits_{i=6}^n \Delta x [f(a + i \Delta x)]\)
\(\displaystyle \Delta x = \dfrac{b - a}{n}\)
\(\displaystyle \Delta x = \dfrac{\dfrac{3\pi}{2} - 0}{6} = \dfrac{\pi}{4}\)
\(\displaystyle n = 6\)
\(\displaystyle \sum\limits_{i=6}^n (\dfrac{\pi}{4}) [[f(0 +(1)(\dfrac{\pi}{4})] + [f(0 +(2)(\dfrac{\pi}{4})] + [f(0 +(3)(\dfrac{\pi}{4})] + [f(0 +(4)(\dfrac{\pi}{4})] + [f(0 + (5)(\dfrac{\pi}{4})] + [f(0 + ( 6)(\dfrac{\pi}{4})]]\)
\(\displaystyle \sum\limits_{i=6}^n (\dfrac{\pi}{4}) [[f(0 + (\dfrac{\pi}{4})] + [f(0 + (\dfrac{2\pi}{4})] + [f(0 + (\dfrac{3\pi}{4})] + [f(0 + (\dfrac{4\pi}{4})] + [f(0 + (\dfrac{5\pi}{4})] + [f(0 + (\dfrac{6\pi}{4})]]\)
\(\displaystyle \sum\limits_{i=6}^n (\dfrac{\pi}{4}) [[f(0 + (\dfrac{\pi}{4})] + [f(0 + (\dfrac{\pi}{2})] + [f(0 + (\dfrac{3\pi}{4})] + [f(0 + \pi)] + [f(0 + (\dfrac{5\pi}{4})] + [f(0 + (\dfrac{3\pi}{2})]]\)
\(\displaystyle \sum\limits_{i=6}^n (\dfrac{\pi}{4}) [[f((\dfrac{\pi}{4})] + [f( (\dfrac{\pi}{2})] + [f( (\dfrac{3\pi}{4})] + [f( \pi)] + [f((\dfrac{5\pi}{4})] + [f((\dfrac{3\pi}{2})]]\)
\(\displaystyle \sum\limits_{i=6}^n (\dfrac{\pi}{4}) [[4 \sin ((\dfrac{\pi}{4})] + [4 \sin ( (\dfrac{\pi}{2})] + [4 \sin ( (\dfrac{3\pi}{4})] + [4 \sin ( \pi)] + [4 \sin ((\dfrac{5\pi}{4})] + [4 \sin ((\dfrac{3\pi}{2})]]\)
\(\displaystyle \sum\limits_{i=6}^n (\dfrac{\pi}{4}) [[4(.7071)] + [4(1)] + [4(.7071)] + [4 (0)] + [4 (-.7071)] + [4 (-1)]]\)
\(\displaystyle \sum\limits_{i=6}^n (\dfrac{\pi}{4}) [[2.8284] + [4] + [2.8284] + [0] + [-2.8284] + [-4]]\)
\(\displaystyle \sum\limits_{i=6}^n (.7854) [[2.8284] + [4] + [2.8284] + [0] + [-2.8284] + [-4]]\)
\(\displaystyle \sum\limits_{i=6}^n [2.2214] + [3.1416] + [2.2214] + [0] + [-2.2214] + [-3.1416]\) - On the right track?
\(\displaystyle \sum\limits_{i=6}^n \Delta x [f(a + i \Delta x)]\)
\(\displaystyle \Delta x = \dfrac{b - a}{n}\)
\(\displaystyle \Delta x = \dfrac{\dfrac{3\pi}{2} - 0}{6} = \dfrac{\pi}{4}\)
\(\displaystyle n = 6\)
\(\displaystyle \sum\limits_{i=6}^n (\dfrac{\pi}{4}) [[f(0 +(1)(\dfrac{\pi}{4})] + [f(0 +(2)(\dfrac{\pi}{4})] + [f(0 +(3)(\dfrac{\pi}{4})] + [f(0 +(4)(\dfrac{\pi}{4})] + [f(0 + (5)(\dfrac{\pi}{4})] + [f(0 + ( 6)(\dfrac{\pi}{4})]]\)
\(\displaystyle \sum\limits_{i=6}^n (\dfrac{\pi}{4}) [[f(0 + (\dfrac{\pi}{4})] + [f(0 + (\dfrac{2\pi}{4})] + [f(0 + (\dfrac{3\pi}{4})] + [f(0 + (\dfrac{4\pi}{4})] + [f(0 + (\dfrac{5\pi}{4})] + [f(0 + (\dfrac{6\pi}{4})]]\)
\(\displaystyle \sum\limits_{i=6}^n (\dfrac{\pi}{4}) [[f(0 + (\dfrac{\pi}{4})] + [f(0 + (\dfrac{\pi}{2})] + [f(0 + (\dfrac{3\pi}{4})] + [f(0 + \pi)] + [f(0 + (\dfrac{5\pi}{4})] + [f(0 + (\dfrac{3\pi}{2})]]\)
\(\displaystyle \sum\limits_{i=6}^n (\dfrac{\pi}{4}) [[f((\dfrac{\pi}{4})] + [f( (\dfrac{\pi}{2})] + [f( (\dfrac{3\pi}{4})] + [f( \pi)] + [f((\dfrac{5\pi}{4})] + [f((\dfrac{3\pi}{2})]]\)
\(\displaystyle \sum\limits_{i=6}^n (\dfrac{\pi}{4}) [[4 \sin ((\dfrac{\pi}{4})] + [4 \sin ( (\dfrac{\pi}{2})] + [4 \sin ( (\dfrac{3\pi}{4})] + [4 \sin ( \pi)] + [4 \sin ((\dfrac{5\pi}{4})] + [4 \sin ((\dfrac{3\pi}{2})]]\)
\(\displaystyle \sum\limits_{i=6}^n (\dfrac{\pi}{4}) [[4(.7071)] + [4(1)] + [4(.7071)] + [4 (0)] + [4 (-.7071)] + [4 (-1)]]\)
\(\displaystyle \sum\limits_{i=6}^n (\dfrac{\pi}{4}) [[2.8284] + [4] + [2.8284] + [0] + [-2.8284] + [-4]]\)
\(\displaystyle \sum\limits_{i=6}^n (.7854) [[2.8284] + [4] + [2.8284] + [0] + [-2.8284] + [-4]]\)
\(\displaystyle \sum\limits_{i=6}^n [2.2214] + [3.1416] + [2.2214] + [0] + [-2.2214] + [-3.1416]\) - On the right track?
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