England_MAT_265_ONLINE_A_Spring_2020.Section_5.2.pdf -...

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England MAT 265 ONLINE A Spring 2020 Assignment Section 5.2 1. (1 point) Consider the integral Z 7 4 ( 4 x 2 + 2 x + 6 ) dx (a) Find the Riemann sum for this integral using right end- points and n = 3. R 3 = (b) Find the Riemann sum for this same integral, using left end- points and n = 3. L 3 = Solution: We have Δ x = 7 - 4 3 = 1 and x 0 = 4 , x 1 = 5 , x 2 = 6 , x 3 = 7. Let f ( x ) = 4 x 2 + 2 x + 6. (a) R 3 = Δ x · [ f ( x 1 )+ f ( x 2 )+ f ( x 3 )] = 1 [ f ( 5 )+ f ( 6 )+ f ( 7 )] = 1 [ 116 + 162 + 216 ] = 494 (b) L 3 = Δ x · [ f ( x 0 )+ f ( x 1 )+ f ( x 2 )] = 1 [ f ( 4 )+ f ( 5 )+ f ( 6 )] = 1 [ 78 + 116 + 162 ] = 356 Correct Answers: 494 356 2. (1 point) Consider the integral Z 9 5 3 x + 2 dx (a) Find the Riemann sum for this integral using right end- points and n = 4. (b) Find the Riemann sum for this same integral, using left end- points and n = 4 Solution: We have Δ x = 9 - 5 3 = 1 and x 0 = 5 , x 1 = 6 , x 2 = 7 , x 3 = 8 , x 4 = 9. Let f ( x ) = 3 x + 2. (a) R 4 = Δ x · [ f ( x 1 )+ f ( x 2 )+ f ( x 3 )+ f ( x 4 )] = 1 [ f ( 6 )+ f ( 7 )+ f ( 8 )+ f ( 9 )] = 1 5 2 + 17 7 + 19 8 + 7 3 = 1619 168 (b) L 4 = Δ x · [ f ( x 0 )+ f ( x 1 )+ f ( x 2 )+ f ( x 3 )] = 1 [ f ( 5 )+ f ( 6 )+ f ( 7 )+ f ( 8 )] = 1 13 5 + 5 2 + 17 7 + 19 8 = 2773 280 Correct Answers: 9.63690476190476 9.90357142857143 3. (1 point) Use the Midpoint Rule to approximate Z 3 . 5 - 2 . 5 x 3 dx with n = 6. Solution: SOLUTION We have Δ x = 3 . 5 + 2 . 5 6 = 1 and x 0 = - 2 . 5 , x 1 = - 1 . 5 ,..., x 6 = 3 . 5. The midpoints are then ¯ x 1 = - 2 , ¯ x 2 = - 1 ,..., ¯ x 6 = 3 Let f ( x ) = x 3 . The midpoint approximation to the integral is 1 · ( f ( - 2 )+ f ( - 1 )+ .. + f ( 3 )) = ( - 2 ) 3 +( - 1 ) 3 + ... +( 3 ) 3 = 27 Correct Answers: 27 4.

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