Question

use the Midpoint Rule with n = 4 to approximate the area of the region bounded by the graph of the function and the x -axis over the given interval.

Answer 1

To use the Midpoint Rule, we need to divide the interval [0, 4] into 4 equal intervals. This give us 4 intervals of width 1 unit, which are the base of the rectangles.

The intervals are:

[0, 1], [1, 2], [2, 3], [3, 4]

And find the midpoint of each interval:

[0, 1] : 0.5

[1, 2] : 1.5

[2, 3] : 2.5

[3, 4] : 3.5

Now, we need to evaluate the function on each midpoint of each interval, which will act as the height of each rectangle:

`\begin{gathered} f(0.5)=0.5^2+4\cdot0.5=(9)/(4) \\ . \\ f(1.5)=1.5^2+4\cdot1.5=(27)/(4) \\ . \\ f(2.5)=2.5^2+4\cdot2.5=(65)/(4) \\ . \\ f(3.5)=3.5^2+4\cdot3.5=(104)/(4) \end{gathered}`

And now, we calculate the area of each rectangle:

`\begin{gathered} (9)/(4)\cdot1=(9)/(4) \\ . \\ (27)/(4)\cdot1=(27)/(4) \\ . \\ (65)/(4)\cdot1=(65)/(4) \\ . \\ (105)/(4)\cdot1=(105)/(4) \end{gathered}`

Finally, we add all the areas and get the midpoint approximation:

`(9+27+65+105)/(4)=(206)/(4)=(103)/(2)=51.5`

By the Midpoint Rule approximation, the area under the curve of f(x) in the interval [0, 4] is 51.5