Functions, delta in summation notation?

Question

Here's what I'm given. Hope I can type it clearly enough.

5
E(for sigma) f(x sub i) delta(x)
i=1

(a) f(x) = 2/3x+1

(b) delta x = 0.5

What do you do here? Many thanks!

Answer 1

5
Σ (f(x(sub i)) δx) (with f(x) = 2/3x+1 and δx = 0.5)
i=1

= f(x(sub1))*0.5 + f(x(sub2))*0.5 + f(x(sub3))*0.5 + f(x(sub4))*0.5 + f(x(sub5))*0.5

= (2/3x(sub1) +1)0.5 + (2/3x(sub2) +1)0.5 + (2/3x(sub3) +1)0.5 + (2/3x(sub4) +1)0.5 + (2/3x(sub5) +1)0.5

Answer 2

You're adding up little rectangle area with height f(x sub i) and width delta x. This is an approximation to the Riemann integral.

When you do the sum, note that the delta x is constant and so can be factored out of the sum:

Sum = (delta x)*(sum over i f(x sub i). Make a table:

i, x sub i, f(x sub i)
-----------------------

enter values...


------------------------

sum the last column and multiply by delta x. This gives you the approximate area under the function.

Answer 3

differentiate f(x).then put the value of delta x and the integrate the entire thing putting it equal to 1/5.