Find the absolute value of the resulting error if the value of the integral from 0 to 3 of x cubed, dx is estimated with 3 inscribed rectangles of equal width.

Question

a) 87.75

b) 15.75

c) 11.25

d) 6.75

Answer 1

The actual integral evaluates to (x^4)/4 from 0 to 3 = 20.25.

 

Graph f(x)=x-cubed.

If the rectangles are INSCRIBED,  I think that means they are fully under the curve, with the left vertical edge of each rectangle touching the graph of x-cubed.  

 

Each rectangle is 1 unit wide.

The rectangle from 0 to 1 has an area of 0   

        [f(0) = 0, so Area = lw = (0)(1) = 0]

The rectangle from 1 to 2 has an area of 1   

        [f(1) = 1, so Area = lw = (1)(1) = 1]

The rectangle from 2 to 3 has an area of 8

        [f(2) = 8, so Area = lw = (8)(1) = 8]

 

So the estimated area is the sum of these three rectangles, so it is

0 + 1 + 8 = 9

 

The error is 9 - 20.25 = -11.25, an UNDERestimate (all of the rectangles lie below the graph, so do not contain all of the area under the x-cubed curve.)

 

The abs val of the error is 11.25