Use the formula to calculate the sum for n = 100, n = 500, and n = 1000. n = 100

Question

Use the formula to calculate the sum for n = 100, n = 500, and n = 1000. n = 100 n = 500 n = 1000 Write out the sigma notation for the Riemann sum described in such a way that the only letter which appears in the general term of the sum is k. Don't calculate the value of the sum; just write down in sigma notation. f(x) =4x^2, [a, b] = [0, 4], left sum, n = 4 sigma^4_k = 1 If integral^4_1 g(x) dx - 9 and integral^7_4 g(x) dx = 6, then find the value of the definite integral. (If not possible, enter IMPOSSIBLE.) integral^7_-1(5g(x) + 5x) dx Consider the following definite integral. Integral^6_1 (6 - x) dx

Explanation / Answer

1)f(x)=4x2 ,a=0,b=4,n=4

=>x=(b-a)/n =(4-0)/4 =1

left end points are x=0,1,2,3

left sum =x[f(0)+f(1)+f(2)+f(3)]

left sum =1*[(4*02)+(4*12)+(4*22)+(4*32)]

left sum =4[k=1]1 *(4*(k-1)2)

left sum =4[k=1] (4k2-8k+4)

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2)

[-1 to 7] (5g(x) +5x)dx

=(5[-1 to 7] g(x)dx)+ [-1 to 7] (5x)dx

=(5([-1 to 4] g(x)dx+[4 to 7] g(x)dx))+ [-1 to 7] (5x)dx

=(5(9+6))+ [-1 to 7] ((5/2)x2)

=(5*15)+ ((5/2)[72-(-1)2])

=75+ ((5/2)*48)

=75+120

=195

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