Questions: Consider the figures below. Figure A Figure B (1) (a) Use the rectangles in each graph to approximate the area of the region bounded by y=sin(x), y=0, x=0, and x=π. (Round your answer to three decimal places.) Figure A Figure B

Consider the figures below.

Figure A

Figure B
(1)
(a) Use the rectangles in each graph to approximate the area of the region bounded by y=sin(x), y=0, x=0, and x=π. (Round your answer to three decimal places.)

Figure A
Figure B

Transcript text: Consider the figures below. Figure A Figure $B$ (1) (a) Use the rectangles in each graph to approximate the area of the region bounded by $y=\sin (x), y=0, x=0$, and $x=\pi$. (Round your answer to three decimal places.) Figure A $\square$ Figure B $\square$

Solution

Solution Steps

Step 1: Identify the function and boundaries

The function given is $ y = \sin(x) $ and the boundaries are $ y = 0 $, $ x = 0 $, and $ x = \pi $.

Step 2: Determine the width of each rectangle

For both figures, the interval from $ 0 $ to $ \pi $ is divided into 4 equal parts. Therefore, the width of each rectangle is: \[ \Delta x = \frac{\pi - 0}{4} = \frac{\pi}{4} \]

Step 3: Calculate the height of each rectangle for Figure A

For Figure A, the heights of the rectangles are determined by the function values at the left endpoints of each subinterval:

$ x = 0 $: $ \sin(0) = 0 $
$ x = \frac{\pi}{4} $: $ \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} $
$ x = \frac{\pi}{2} $: $ \sin\left(\frac{\pi}{2}\right) = 1 $
$ x = \frac{3\pi}{4} $: $ \sin\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2} $

Step 4: Calculate the area of each rectangle for Figure A

The area of each rectangle is given by $ \text{height} \times \text{width} $:

Area 1: $ 0 \times \frac{\pi}{4} = 0 $
Area 2: $ \frac{\sqrt{2}}{2} \times \frac{\pi}{4} = \frac{\pi\sqrt{2}}{8} $
Area 3: $ 1 \times \frac{\pi}{4} = \frac{\pi}{4} $
Area 4: $ \frac{\sqrt{2}}{2} \times \frac{\pi}{4} = \frac{\pi\sqrt{2}}{8} $

Step 5: Sum the areas for Figure A

\[ \text{Total Area for Figure A} = 0 + \frac{\pi\sqrt{2}}{8} + \frac{\pi}{4} + \frac{\pi\sqrt{2}}{8} = \frac{\pi\sqrt{2}}{4} + \frac{\pi}{4} \approx 1.896 \]

Step 6: Calculate the height of each rectangle for Figure B

For Figure B, the heights of the rectangles are determined by the function values at the midpoints of each subinterval: