Math Is Fun Forum

If anyone can help it would be great!

AREAS

For the following functions, you will estimate the area underneath the curves between x=1 and x=3. You will be using rectangles to help you estimate this area.

Each rectangle you use in your estimation will have equal widths. The number of rectangles use will be given. EX. USING 8 RECTANGLES IN THE INTERVAL FROM X=1 TO X=5 WILL RESULT IN THE WIDTHS BEING 0.5 IN LENGTH. The base of each rectangle will be the x-axis and the upper left corner of the rectangle will make contact with the curve. The upper right corner will either be above or below the curve. The left had will determine the height.

Procedure for estimating an area. Function: y=2x+1 from x=1 to x=3 using 4 rectangles. Widths of each rectangle will be 0.5. Height of the first rectangle is determined by using x=1. Height of second rectangle is determined by using x=1.5. Height of the third rectangle is determined by using x=2. And the height of the fourth rectangle is determined by using x=2.5. Add all the areas of each rectangle to get estimation. Notice that this estimation is an underestimate based on all rectangles being drawn below curve, leaving some "empty space" not accounted for.

Find the approximate areas for each function from x=1 to x=3 using 8 rectangles of equal width.
1. y=-x²+4x
2. y=x²-4x+5
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7. Approximate the area of the following semicricle y= √ -x²+6x. Use 12 rectangles of equal width. Then, find actual area of semicircle and compare with the estimation.

So what i figured out is that for #1 each rectangle will have a width of 0.25. Then do I plug in the x into y=-x²+4x? Like y=-(1)²+4(1) which would give me y=3. Then I multiply 0.25(3) and get 0.75 for the area of the first rectangle out of 8.... so then just add all of the rectangles together?

Let me know if I am doing it right... I don't know how to do #7 so if someone could explain that would be great!