Chapter 12.5, Problem 31E

To determine

To find:Approximating the area of region bounded by the graph of and the x-axis over the specified interval using the indicated numbers of rectangles of equal width and the exact area as .

Answer to Problem 31E

Approximating the area of region with the number n of rectangles is

n
Approximate area

Explanation of Solution

Given information:

The givenfunction is,

  

Given intervals

And, rectangles are

Concept used:

The area of a region;

Let becontinuousandnonnegativeontheinterval Thearea Aoftheregionboundedbythegraphof ,the ,andtheverticallines and isgivenby

  

The givenfunctionis,

  

Thearea of the region bounded by the graph of and the x-axis between and

Thedimensionsoftherectangles;

  

And,

  

Given function;

  

Approximate the areas as the sum of the areas of rectangles.

  

Applying Summationformulasandproperties.

  

Now,

  

Obtain a more accurate approximation of the area of the region by increasing the number of rectangles.

  

Take,

  

  

In this way,

The exact area by taking the limit as approaches

  

Therefore, the exact area is

Now, values of

Finally, toapproximating the area of region by using values inthe following table.

n
Approximate area