The estimated areas are listed below:
- The area of the function is approximately 0.806.
- The area of the function is approximately 2.088.
- The area of the function is approximately 0.696.
How to approximate the area of a given region with a given number of subintervals
Right Riemann sum based on trapezoids offer a compact method that combines upper and lower sums, which is defined below:
[tex]A = \frac{b-a}{2\cdot n}\cdot \sum_{i = 0}^{n-1} [f(x_{i})+f(x_{i+1})][/tex] (1)
Where:
- b - Upper bound
- a - Lower bound
- n - Number of subintervals
- i - Subinterval index
In this question we must apply right Riemann sums to estimate the areas of functions on given intervals. Now we proceed to solve each case below:
Case I ([tex]f(x) = \sqrt{x}[/tex], [tex]a = 0[/tex], [tex]b = 1[/tex], [tex]n = 4[/tex])
[tex]A = \frac{1-0}{2\cdot (4)} \cdot [(0+\sqrt{0.5})+(\sqrt{0.5} + 1)+(1+\sqrt{1.5})+(\sqrt{1.5}+2)][/tex]
A ≈ 0.806
The area of the function is approximately 0.806. [tex]\blacksquare[/tex]
Case II ([tex]f(x) = 4\cdot e^{-x}[/tex], [tex]a = 0[/tex], [tex]b = 2[/tex], [tex]n = 4[/tex])
[tex]A = \frac{2-0}{2\cdot (4)}\cdot [(4+1.472)+(1.472+0.541)+(0.541+0.199)+(0.199-0.073)][/tex]
A ≈ 2.088
The area of the function is approximately 2.088. [tex]\blacksquare[/tex]
Case III ([tex]f(x) = \frac{1}{x}[/tex], [tex]a = 1[/tex], [tex]b = 2[/tex], [tex]n = 5[/tex])
[tex]A = \frac{2-1}{2\cdot (5)} \cdot [(1+0.833)+(0.833+0.714)+(0.714+0.625)+(0.625+0.556)+(0.556+0.5)][/tex]
A ≈ 0.696
The area of the function is approximately 0.696. [tex]\blacksquare[/tex]
To learn more on Riemann sums, we kindly invite to check this verified question: https://brainly.com/question/21847158
