To find: The expression for area under the curve .
The area under the curve is expressed as .
The function is .
The upper limit is and lower limit is .
The area A of the region (S) under the graph f of a continuous function is the sum of area of the approximating rectangles is given by,
Find the width of the interval using the relation:
Here, the upper limit is b, the lower limit is a, and the number of rectangles is n.
Substitute 1 for b and 0 for a in Equation (2).
Find the value of using the relation:
Substitute 0 for a and for in Equation (3).
Use definition 2 to obtain the expression for area under the curve.
Substitute for and for in equation (1).
Substitute for in equation (4).
Therefore, the expression for the area under the curve is .
To evaluate: The limit .
The value of the limit is .
The general expression of sum of cubes for the first n integers is shown below:
Rearrange the limit function as shown below.
Apply the general expression of sum of cubes in equation (1).
On further simplification,
Therefore, the value of the limit is .
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