Approximate area under y=5/x from x=1 to x=5 using rectangles
Check the final answer first, then review the worked steps.
Problem
Approximate area under y=5/x from x=1 to x=5 using rectangles
Answer
Continually increase the number of rectangles.
Step-by-step solution
- Understand the goal: The problem asks to approximate the area under the curve $y = \frac{5}{x}$ from $x=1$ to $x=5$ using Riemann sums (rectangles).
- Analyze Figure A: In Figure A, there are 4 rectangles, each with a width of $\Delta x = \frac{5-1}{4} = 1$. The heights are determined by the left endpoints of each interval: $f(1)=5$, $f(2)=2.5$, $f(3)=1.667$, and $f(4)=1.25$. The total area is $1(5 + 2.5 + 1.667 + 1.25) = 10.417$.
- Analyze Figure B: In Figure B, there are 8 rectangles, each with a width of $\Delta x = \frac{5-1}{8} = 0.5$. The heights are determined by the left endpoints: $f(1)=5, f(1.5)=3.333, f(2)=2.5, f(2.5)=2, f(3)=1.667, f(3.5)=1.429, f(4)=1.25, f(4.5)=1.111$. The total area is $0.5(5 + 3.333 + 2.5 + 2 + 1.667 + 1.429 + 1.25 + 1.111) = 9.145$.
- Conclusion on accuracy: As the number of rectangles increases, the width of each rectangle ($\Delta x$) decreases, and the sum of the areas of the rectangles becomes a better approximation of the actual area under the curve (the definite integral). Therefore, to obtain a more accurate approximation, one should continually increase the number of rectangles.