Finding Area by the Limit Definition In Exercises 57-62, use the limit process to find the area of the region bounded by the graph of the function and the y-axis over the given y-interval. Sketch the region.
To calculate: The area of the region bounded by the function , in the interval and the -axis.
Given: The provided function is,
, in the interval .
Formula used: The sum of squares of first natural numbers is given by the formula,
The sum of first natural numbers is given by the formula,
Sum of a constant times is written as,
The sum of cubes of first natural numbers is given by the formula,
Using upper endpoints area is,
Here, is the upper endpoints.
Calculation: The provided function is,
We will differentiate with respect to and obtain ,
Now, equate to zero to get,
So, the slope of the function w.r.t -axis is zero at one point. Check where the curve is approach by put and .
So, is approach to .
That means the curve is approaching to as .
Therefore, the required graph with the region is shown below,
As the given function is continuous and non-negative in the interval . So, by the partition of interval into subintervals each of equal width.
Therefore, considering the width of each rectangle to be then,
The width of each rectangle is,
The area can be calculated by the use of lower endpoints or upper endpoints . For this problem the upper endpoints are convenient.
Now, find upper endpoints .
So, the upper endpoint is .
Therefore, apply the formula to find the area of the region,
Now, substitute the values to find the area of the region
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