Question

question 4
for a function ( f ) that is strictly increasing on ( a, b ), a right riemann sum with ( n = 4 ) that approximates the area under ( f ) on ( a, b ) is which of the following:
(hint: sketch a graph and draw ( n ) rectangles.)
the exact area
cannot be determined
underestimation
overestimation

Explanation:

Step1: Understand Right - Riemann Sum

For a function \(y = f(x)\) on the interval \([a,b]\) with \(n\) sub - intervals \(\Delta x=\frac{b - a}{n}\), and \(x_i=a + i\Delta x\) for \(i = 0,1,\cdots,n\). The right - Riemann sum is \(R_n=\sum_{i = 1}^{n}f(x_i)\Delta x\).

Step2: Analyze the function \(y = f(x)\) (strictly increasing)

Since \(f(x)\) is strictly increasing on \([a,b]\), for each sub - interval \([x_{i-1},x_i]\) (\(i = 1,\cdots,n\)), \(f(x_i)>f(x)\) for all \(x\in[x_{i - 1},x_i]\).
The area of the rectangle in the right - Riemann sum with height \(f(x_i)\) and width \(\Delta x\) overestimates the area under the curve \(y = f(x)\) on the sub - interval \([x_{i-1},x_i]\).

Answer:

Overestimation