Work through the following steps to find the area under f(x) = 2x + 4 on [0, 5], using the limit definition.
a) We know that a = 0 and b = 5.
b) Using n subintervals, Δx = (b - a)/n.
c) Assume that the sample points in each interval are right endpoints. Find the following sample points:
x1 = 0 + (5/n)
x2 = 0 + (10/n)
x3 = 0 + (15/n)
In general, the ith sample point is xi = 0 + (5i/n).
Note: your answer will be an expression in terms of i and n.
d) Now find the sum of the areas of n approximating rectangles.
Σf(xi)Δx = Σ(2xi + 4)Δx
e) Finally, find the exact area by letting the number of rectangles approach infinity.
lim(n→∞) Σ(2xi + 4)Δx