callculus3

Question

Use two rectangles of equal width to estimate the area between the graph of f(x) = x + cos(πx) and the x-axis on the interval [2, 6]. Evaluate the function at the mid-point of each rectangle to find each height.  

16

3

12

6

Answer 1

We find the base of the rectangles by taking the difference between the interval endpoints, and dividing by 2.

 

Base of rectangle = (6 - 2) / 2

                          = 2

 

The area of the first rectangle:

 

(4 - 2)f(4) = 2[4 + cos(4π)]

 

 

The area the second triangle:

 

(6 - 4)f(6) = 2[6 + cos(6π)]

 

 

Now just compute the two areas and combined them.  That will give you the estimated under the curve.

 

 

To evaluate the midpoint of each rectangle, we take the midpoint of the base lengths of each rectangle.   This midpoint is the x value.  Then evaluate the function at that x value.

 

 

The midpoint of the first rectangle is x=3.  Evaluate f(3).

The midpoint of the second rectangle is x=5.  Evaluate f(5).