a Express the area under the curve y=x4+5×2+x from 2 to 7 as a limit. b Use a computer algebra system to evaluate the sum in part a. c Use a computer algebra system to find the exact area by evaluating the limit of the expression in part b.
The area under the curve
The area of the region that lies under the graph of the continuous function is the limit of the sum of the areas of approximating rectangles
The width of the interval is , so the width of each strip is
where and . The right endpoints of the subintervals are .
It is given that
Now find out
Let be the area under the curve , then
We have , and So,
This is the area under the given curve.
The sum in the expression of the area under the curve
From part a),
Now simplify only the summation part.
By using the command in Mathematica system, so the sum is,
The exact area by evaluating the limit
From part a) and b),
Substitute for the summation part, and evaluate the limit.