This problem is a reprise of problem 5 with 1.1 replaced by 1.001 Compute an app

Question

This problem is a reprise of problem 5 with 1.1 replaced by 1.001 Compute an approximation to integral^9_1 1/x dx, which gives the area under y = 1/x for 1 lessthanorequalto x lessthanorequalto 9, using a modified Riemann sum with the (NOT equally spaced) partition 1, 1.001, 1.001^2, 1.001^3, .. ., 1.001^N, 9 and left hand endpoints EXCEPT neglecting the area of the last rectangle. Here N denotes the largest possible power which fits in the interval 1 lessthanorequalto x lessthanorequalto 9. Please note that the problem is NOT asking for the value of integral^9_1 1/x dx. Rather it is asking for the EXACT values of the areas of approximating rectangles and for the EXACT value of the sum of the areas of the rectangles. Calculator approximations (no matter how accurate) will NOT be accepted. Do the calculations by hand using fractions (until you notice the pattern in the areas). The number of approximating rectangles is: N = The area of the first rectangle = The area of the second rectangle = The area of the third rectangle = The sum of the areas of the N rectangles =

Explanation / Answer

the number of approximating rectangles = infinity as there could be infinite sub interval of x E [1,9]

The area of the first rectangle = length*width = x*f(x) = (1.001-1)*f(1.001-1) = (1.001-1)*1/(1.001-1) =1 units square

THe area of the second rectangle = (1.001^2-1.001)*1/(1.001^2-1.001) = 1 units square

The area of the third rectangle = (1.001^3-1.001^2)*1/(1.001^3-1.001^2) = 1 units square

The sum of the areas of the N rectangles is = 1+1+1+1+..........1 = N units square

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