Draw the graph of ƒ(x) = sin(x2/2) in the viewing rectangle [0, 1] by [0, 0.5] and let width=.
(a) Use the graph to decide whether L2, R2, M2, and T2 underestimate or overestimate I.
(b) For any value of , list the numbers Ln, Rn, Mn, Tn, and I in increasing order.
(c) Compute L5, R5, M5, and T5. From the graph, which do you think gives the best estimate of I?
The Correct Answer and Explanation is :
(a) Overestimation and Underestimation Analysis
To determine whether the left sum (L_2), right sum (R_2), midpoint sum (M_2), and trapezoidal sum (T_2) overestimate or underestimate the integral:
- Left sum ( L_n ): Uses left endpoints; since ( f(x) ) is increasing on ([0,1]), ( L_n ) underestimates the integral.
- Right sum ( R_n ): Uses right endpoints; since ( f(x) ) is increasing, ( R_n ) overestimates the integral.
- Midpoint sum ( M_n ): Uses midpoints; provides a more balanced approximation.
- Trapezoidal sum ( T_n ): Averages ( L_n ) and ( R_n ); typically more accurate.
Thus:
- ( L_2 ) underestimates.
- ( R_2 ) overestimates.
- ( M_2 ) provides a more accurate estimate.
- ( T_2 ) refines the estimate between ( L_2 ) and ( R_2 ).
(b) Order of Values for Any ( n )
For any ( n ), the relationship follows:
[
L_n \leq M_n \leq I \leq T_n \leq R_n
]
where ( I ) represents the actual integral value.
(c) Computation of ( L_5, R_5, M_5, T_5 )
We will compute these sums numerically.
Results for ( n = 5 ):
- ( L_5 = 0.1187 )
- ( R_5 = 0.2146 )
- ( M_5 = 0.1622 )
- ( T_5 = 0.1666 )
Best Estimate of ( I ):
From the function’s behavior and past numerical analysis, the midpoint sum ( M_n ) typically provides the most accurate approximation for definite integrals because it accounts for concavity better than left or right sums. However, the trapezoidal rule ( T_n ), being an average of ( L_n ) and ( R_n ), often performs well too.
Given the results, ( M_5 ) (0.1622) and ( T_5 ) (0.1666) are close, with ( T_5 ) likely being a slightly better approximation of ( I ). The trapezoidal rule generally balances out the overestimation from ( R_n ) and the underestimation from ( L_n ), making it a preferred choice.
Explanation (300 Words)
Numerical integration approximations like Riemann sums and the trapezoidal rule are essential for estimating definite integrals when an exact antiderivative is difficult to compute. Given ( f(x) = \sin(x^2/2) ), which is increasing on ([0,1]), we analyze different sum approximations.
The left Riemann sum (( L_n )) underestimates the integral because it always takes function values at the beginning of each subinterval, missing the increase in function value. The right Riemann sum (( R_n )) overestimates because it takes function values at the right endpoints, capturing more of the higher function values.
The midpoint rule (( M_n )) provides an improved estimate because it uses values at the center of each subinterval, better averaging over the curve’s behavior. The trapezoidal rule (( T_n )) is even more refined, averaging ( L_n ) and ( R_n ) to provide a balanced approximation.
For ( n = 5 ), we find:
- ( L_5 = 0.1187 ) (underestimate)
- ( R_5 = 0.2146 ) (overestimate)
- ( M_5 = 0.1622 ) (closer to actual integral)
- ( T_5 = 0.1666 ) (best overall estimate)
Between ( M_5 ) and ( T_5 ), the trapezoidal rule provides the best estimate of the integral because it balances overestimation and underestimation. If a higher accuracy is needed, increasing ( n ) (subintervals) further refines the approximation.