CHAPTER 5
Integration EXERCISE SET 5.1 n−1 1 2 , 1; using right endpoints 1. Endpoints 0, , , . . . , n n n 1 2 n−1 1 An = + + ··· + +1 n n n n n An
2 0.853553
5 0.749739
10 0.710509
50 0.676095
100 0.671463
n−1 1 2 2. Endpoints 0, , , . . . , , 1; using right endpoints n n n n 1 1 n n n + ··· + + + + An = 2n − 1 2 n n+1 n+2 n+3 n An
2 0.583333
3. Endpoints 0,
5 0.645635
10 0.668771
50 0.688172
100 0.690653
(n − 1)π π 2π , ,..., , π; using right endpoints n n n
An = [sin(π/n) + sin(2π/n) + · · · + sin(π(n − 1)/n) + sin π] n An
2 1.57080
4. Endpoints 0,
5 10 50 1.93376 1.98352 1.99935
π n
100 1.99984
(n − 1)π π π 2π , ,..., , ; using right endpoints 2n 2n 2n 2
An = [cos(π/2n) + cos(2π/2n) + · · · + cos((n − 1)π/2n) + cos(π/2)] n An
2 0.555359
5 0.834683
10 0.919405
50 0.984204
π 2n
100 0.992120
n+1 n+2 2n − 1 5. Endpoints 1, , ,..., , 2; using right endpoints n n n n n n 1 1 + + ··· + + An = n+1 n+2 2n − 1 2 n n An
2 0.583333
5 0.645635
10 0.668771
50 0.688172
100 0.690653
π (n − 1)π π π π π π 2π ,...,− + , ; using right endpoints 6. Endpoints − , − + , − + 2 2 n 2 n 2 n 2 π π π π π 2π π (n − 1)π + cos − + + · · · + cos − + + cos An = cos − + 2 n 2 n 2 n 2 n n An
2 1.99985
5 10 50 1.93376 1.98352 1.99936
100 1.99985
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