(a) Use the definition to find an expression for the area under the curve y = x3 from 0 to 1 as a limit.lim n→∞ ni = 1

The area under the curve y = x^3 from 0 to 1 can be approximated by the sum of the areas of n rectangles under the curve. The width of each rectangle is 1/n and the height is the value of the function at the right endpoint of the rectangle.

The area of the i-th rectangle is (1/n) * ((i/n)^3). The total area under the curve is the sum of the areas of these rectangles, which is the sum from i=1 to n of (1/n) * ((i/n)^3).

This sum can be written as (1/n^4) * (sum from i=1 to n of i^3).

The sum of the cubes of the first n integers is given by the formula (n(n+1)/2)^2.

So, the total area under the curve is (1/n^4) * ((n(n+1)/2)^2).

As n approaches infinity, this expression becomes the limit as n approaches infinity of (1/4) * ((n+1)^2/n^2) = (1/4) * (1 + 2/n + 1/n^2).

As n approaches infinity, the terms 2/n and 1/n^2 approach 0, so the limit is (1/4) * 1 = 1/4.

So, the area under the curve y = x^3 from 0 to 1 is 1/4.