Find the Riemann sum for
f ( x ) = 4 sin x \displaystyle f(x) = 4 \sin x f ( x ) = 4 sin x , over interval
[ 0 , 3 π 2 ] \displaystyle [0, \dfrac{3\pi}{2}] [ 0 , 2 3 π ] with six terms, taking the sample points to be right endpoints. (Round your answers to six decimal places.) Find
R 6 \displaystyle R_{6} R 6 - the sixth right endpoint.
∑ i = 6 n Δ x [ f ( a + i Δ x ) ] \displaystyle \sum\limits_{i=6}^n \Delta x [f(a + i \Delta x)] i = 6 ∑ n Δ x [ f ( a + i Δ x ) ] Δ x = b − a n \displaystyle \Delta x = \dfrac{b - a}{n} Δ x = n b − a Δ x = 3 π 2 − 0 6 = π 4 \displaystyle \Delta x = \dfrac{\dfrac{3\pi}{2} - 0}{6} = \dfrac{\pi}{4} Δ x = 6 2 3 π − 0 = 4 π n = 6 \displaystyle n = 6 n = 6 ∑ i = 6 n ( π 4 ) [ [ f ( 0 + ( 1 ) ( π 4 ) ] + [ f ( 0 + ( 2 ) ( π 4 ) ] + [ f ( 0 + ( 3 ) ( π 4 ) ] + [ f ( 0 + ( 4 ) ( π 4 ) ] + [ f ( 0 + ( 5 ) ( π 4 ) ] + [ f ( 0 + ( 6 ) ( π 4 ) ] ] \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})]] i = 6 ∑ n ( 4 π ) [ [ f ( 0 + ( 1 ) ( 4 π ) ] + [ f ( 0 + ( 2 ) ( 4 π ) ] + [ f ( 0 + ( 3 ) ( 4 π ) ] + [ f ( 0 + ( 4 ) ( 4 π ) ] + [ f ( 0 + ( 5 ) ( 4 π ) ] + [ f ( 0 + ( 6 ) ( 4 π ) ] ] ∑ i = 6 n ( π 4 ) [ [ f ( 0 + ( π 4 ) ] + [ f ( 0 + ( 2 π 4 ) ] + [ f ( 0 + ( 3 π 4 ) ] + [ f ( 0 + ( 4 π 4 ) ] + [ f ( 0 + ( 5 π 4 ) ] + [ f ( 0 + ( 6 π 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})]] i = 6 ∑ n ( 4 π ) [ [ f ( 0 + ( 4 π ) ] + [ f ( 0 + ( 4 2 π ) ] + [ f ( 0 + ( 4 3 π ) ] + [ f ( 0 + ( 4 4 π ) ] + [ f ( 0 + ( 4 5 π ) ] + [ f ( 0 + ( 4 6 π ) ] ] ∑ i = 6 n ( π 4 ) [ [ f ( 0 + ( π 4 ) ] + [ f ( 0 + ( π 2 ) ] + [ f ( 0 + ( 3 π 4 ) ] + [ f ( 0 + π ) ] + [ f ( 0 + ( 5 π 4 ) ] + [ f ( 0 + ( 3 π 2 ) ] ] \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})]] i = 6 ∑ n ( 4 π ) [ [ f ( 0 + ( 4 π ) ] + [ f ( 0 + ( 2 π ) ] + [ f ( 0 + ( 4 3 π ) ] + [ f ( 0 + π ) ] + [ f ( 0 + ( 4 5 π ) ] + [ f ( 0 + ( 2 3 π ) ] ] ∑ i = 6 n ( π 4 ) [ [ f ( ( π 4 ) ] + [ f ( ( π 2 ) ] + [ f ( ( 3 π 4 ) ] + [ f ( π ) ] + [ f ( ( 5 π 4 ) ] + [ f ( ( 3 π 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})]] i = 6 ∑ n ( 4 π ) [ [ f ( ( 4 π ) ] + [ f ( ( 2 π ) ] + [ f ( ( 4 3 π ) ] + [ f ( π ) ] + [ f ( ( 4 5 π ) ] + [ f ( ( 2 3 π ) ] ] ∑ i = 6 n ( π 4 ) [ [ 4 sin ( ( π 4 ) ] + [ 4 sin ( ( π 2 ) ] + [ 4 sin ( ( 3 π 4 ) ] + [ 4 sin ( π ) ] + [ 4 sin ( ( 5 π 4 ) ] + [ 4 sin ( ( 3 π 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})]] i = 6 ∑ n ( 4 π ) [ [ 4 sin ( ( 4 π ) ] + [ 4 sin ( ( 2 π ) ] + [ 4 sin ( ( 4 3 π ) ] + [ 4 sin ( π ) ] + [ 4 sin ( ( 4 5 π ) ] + [ 4 sin ( ( 2 3 π ) ] ] ∑ i = 6 n ( π 4 ) [ [ 4 ( . 7071 ) ] + [ 4 ( 1 ) ] + [ 4 ( . 7071 ) ] + [ 4 ( 0 ) ] + [ 4 ( − . 7071 ) ] + [ 4 ( − 1 ) ] ] \displaystyle \sum\limits_{i=6}^n (\dfrac{\pi}{4}) [[4(.7071)] + [4(1)] + [4(.7071)] + [4 (0)] + [4 (-.7071)] + [4 (-1)]] i = 6 ∑ n ( 4 π ) [ [ 4 ( . 7 0 7 1 ) ] + [ 4 ( 1 ) ] + [ 4 ( . 7 0 7 1 ) ] + [ 4 ( 0 ) ] + [ 4 ( − . 7 0 7 1 ) ] + [ 4 ( − 1 ) ] ] ∑ i = 6 n ( π 4 ) [ [ 2.8284 ] + [ 4 ] + [ 2.8284 ] + [ 0 ] + [ − 2.8284 ] + [ − 4 ] ] \displaystyle \sum\limits_{i=6}^n (\dfrac{\pi}{4}) [[2.8284] + [4] + [2.8284] + [0] + [-2.8284] + [-4]] i = 6 ∑ n ( 4 π ) [ [ 2 . 8 2 8 4 ] + [ 4 ] + [ 2 . 8 2 8 4 ] + [ 0 ] + [ − 2 . 8 2 8 4 ] + [ − 4 ] ] ∑ i = 6 n ( . 7854 ) [ [ 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]] i = 6 ∑ n ( . 7 8 5 4 ) [ [ 2 . 8 2 8 4 ] + [ 4 ] + [ 2 . 8 2 8 4 ] + [ 0 ] + [ − 2 . 8 2 8 4 ] + [ − 4 ] ] ∑ i = 6 n [ 2.2214 ] + [ 3.1416 ] + [ 2.2214 ] + [ 0 ] + [ − 2.2214 ] + [ − 3.1416 ] \displaystyle \sum\limits_{i=6}^n [2.2214] + [3.1416] + [2.2214] + [0] + [-2.2214] + [-3.1416] i = 6 ∑ n [ 2 . 2 2 1 4 ] + [ 3 . 1 4 1 6 ] + [ 2 . 2 2 1 4 ] + [ 0 ] + [ − 2 . 2 2 1 4 ] + [ − 3 . 1 4 1 6 ] - On the right track?
Last edited: Nov 16, 2013
