Chapter 4: Integration
Section 4.1: Area by Riemann Sums
Use Maple to obtain −125n3∑i=1ni2−i⁢n as the right Riemann sum for fx=6+x−x2, −2≤x≤3.
Use Maple to evaluate this Riemann sum to 1256 n2−1n2.
Obtain closed-form expressions for ∑k=1na, ∑k=1nk, and ∑k=1nk2; then use these expressions to show how the right Riemann sum becomes 1256 n2−1n2.
Table 4.1.3(a) lists Maple-computed closed-form expressions for sums of k0=1,k, and k2. The summation symbol is provided by the Expression palette. The Context Panel options Evaluate and Display Inline, and Factor are used to obtain and simplify the results.
∑k=1n1 = n
∑k=1nk = 12⁢n+12−12⁢n−12= factor 12⁢n⁢n+1
∑k=1nk2 = 13⁢n+13−12⁢n+12+16⁢n+16= factor 16⁢n⁢n+1⁢2⁢n+1
Table 4.1.3(a) Closed-form expressions for sums of powers of the index k
Table 4.1.3(b) contains the Maple calculations for the right Riemann sum and its simplifications. The RiemannSum command from the Student Calculus1 package is used to obtain the sum. (This command is also used in the Riemann Sum task templates.)
Initialize and define the function f
Tools≻Load Package: Student Calculus 1
Context Panel: Define Function
fx=6+x−x2→assign as functionf
Obtain the right Riemann sum
Apply the RiemannSum command and press the Enter key.
Context Panel: Evaluate Sum
Context Panel: Simplify≻Simplify
Table 4.1.3(b) Maple's calculation of the right Riemann sum for fx on the interval −2,3
The passage from −125n3∑i=1ni2−i n to 1256n2−1n2 is effected by using the expressions in Table 4.1.5.
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