Question

In Exercises \(53-56,\) use the Midpoint Rule Area \(\approx \sum_{i=1}^{n} f\left(\frac{x_{i}+x_{i-1}}{2}\right) \Delta x\) 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. $$ f(x)=x^{2}+4 x, \quad[0,4] $$

Short answer

The approximate area under the curve is 53 units.

Step-by-step solution

Interpret the interval [0,4]

Firstly, the given interval is [0,4]. This interval can be divided into 4 equal parts because \(n=4\). When these 4 parts are calculated, it results to \(\Delta x = (4 - 0) / 4 = 1 \) . This means that the subintervals are [0,1], [1,2], [2,3] and [3,4].

Applying the Midpoint Rule

On each subinterval, the Midpoint Rule tells to calculate the function at the middle of each interval, multipy it with \(\Delta x\), and finally to sum up all those values. Therefore, calculate for the midpoints which are at \(0.5, 1.5, 2.5\) and \(3.5\). Substituting these midpoints into the function: \(f(0.5)=0.5^2+4 (0.5)=2.25\), \(f(1.5)=1.5^2+4(1.5)=8.25\), \(f(2.5)=2.5^2+4 (2.5)=16.25\), \(f(3.5)=3.5^2 +4 (3.5)=26.25\).

Summing the results

According to the Midpoint Rule, the total sum would be \(\Delta x\) multiplied by the sum of the function values at midpoints. Therefore, the area would be \(Area \approx \Delta x [f(0.5)+f(1.5)+f(2.5)+f(3.5)]= 1(2.25+8.25+16.25+26.25)=53\).