Approximate area under y=sin(x) using rectangles
Check the final answer first, then review the worked steps.
Problem
Approximate area under y=sin(x) using rectangles
Answer
Figure A: 3.142, Figure B: 2.052; The process is improved by 'Continually increase the number of rectangles'.
Step-by-step solution
- Analyze Figure A: The interval $[0, \pi]$ is divided into 2 rectangles of width $\Delta x = \frac{\pi}{2}$. The heights are determined by the function $y = \sin(x)$. For the first rectangle on $[0, \frac{\pi}{2}]$, the height is $\sin(\frac{\pi}{2}) = 1$. For the second rectangle on $[\frac{\pi}{2}, \pi]$, the height is $\sin(\frac{\pi}{2}) = 1$. The total area is $A = (\frac{\pi}{2} \cdot 1) + (\frac{\pi}{2} \cdot 1) = \pi \approx 3.142$.
- Analyze Figure B: The interval $[0, \pi]$ is divided into 4 rectangles of width $\Delta x = \frac{\pi}{4}$. The heights are determined by the function $y = \sin(x)$ at the midpoints of the subintervals: $\frac{\pi}{8}, \frac{3\pi}{8}, \frac{5\pi}{8}, \frac{7\pi}{8}$. The heights are $\sin(\frac{\pi}{8}) \approx 0.3827$, $\sin(\frac{3\pi}{8}) \approx 0.9239$, $\sin(\frac{5\pi}{8}) \approx 0.9239$, and $\sin(\frac{7\pi}{8}) \approx 0.3827$. The total area is $A = \frac{\pi}{4} (0.3827 + 0.9239 + 0.9239 + 0.3827) = \frac{\pi}{4} (2.6132) \approx 2.052$.
- Determine the method for improvement: As the number of rectangles increases, the width of each rectangle decreases, allowing the sum of the areas of the rectangles to better approximate the area under the curve. This is the fundamental concept of the Riemann sum, which converges to the definite integral as the number of rectangles approaches infinity.