Repeat Exercise 25 for ∫₀¹ x³ dx.

Exercise 25

In this exercise, we investigate the value of ∫₀¹ x² dx using larger and larger values of n in the definition of the definite integral.

(a) First let n = 10, so Δx = 0.1. Fill a list on your calculator with values of x² as x goes from 0.1 to 1. (On a TI-84 calculator, use the command seq(X^2,X,.1,1,.1)→L1.)
(b) Sum the values in the list formed in part (a), and multiply by 0.1, to estimate ∫₀¹ x² dx with n = 10. (On a TI-84 calculator, use the command .1* sum(L1).)

(c) Repeat parts (a) and (b) with n = 100.
(d) Repeat parts (a) and (b) with n = 500.
(e) Based on your answers to parts (b) through (d), what do you estimate the value of  ∫₀¹ x² dx to be?