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:
[tex]\begin{gathered} f(0.5)=0.5^2+4\cdot0.5=\frac{9}{4} \\ . \\ f(1.5)=1.5^2+4\cdot1.5=\frac{27}{4} \\ . \\ f(2.5)=2.5^2+4\cdot2.5=\frac{65}{4} \\ . \\ f(3.5)=3.5^2+4\cdot3.5=\frac{104}{4} \end{gathered}[/tex]And now, we calculate the area of each rectangle:
[tex]\begin{gathered} \frac{9}{4}\cdot1=\frac{9}{4} \\ . \\ \frac{27}{4}\cdot1=\frac{27}{4} \\ . \\ \frac{65}{4}\cdot1=\frac{65}{4} \\ . \\ \frac{105}{4}\cdot1=\frac{105}{4} \end{gathered}[/tex]Finally, we add all the areas and get the midpoint approximation:
[tex]\frac{9+27+65+105}{4}=\frac{206}{4}=\frac{103}{2}=51.5[/tex]By the Midpoint Rule approximation, the area under the curve of f(x) in the interval [0, 4] is 51.5
