Evaluate by interpreting it as the area of part of a familiar geometric figure.

Question

lim n to infinity 1/ n ∑ n j = 1 √19 - 19(j/n)^2

Answer 1

The purpose of this exercise is for you to recognize that for any function f(x)

the definite integral from 0 to 1

∫¹₀ f(x)dx = lim (n->∞) [ (1/n) Σⁿ_j=1 f(j/n) ]

You are approximating the integral as a Riemann sum.

You are dividing the interval between 0 and 1 into n subinterval.

The area under the curve of f is approximated by rectangles, one per subinterval.

The width of each rectangle is 1/n the the height is f(j/n).

In your particular example

f(x) = √19 - 19x²

The indefinite integral

F(x) = ∫(√19 - 19x²)dx = x√19 - (19/3)x³ + C

The definite integral between 0 and 1 is then

F(1) - F(0) = √19 - (19/3)