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# Area under a curve

• Find an estimate of the area under the graph of between and above the -axis. Use four left endpoint rectangles.
• Find an estimate of the area under the graph of between and above the -axis. Use four right endpoint rectangles.
• Find an estimate of the area under the graph of between and . Use four left endpoint rectangles.
• Find an estimate of the area under the graph of between and . Use two midpoint rectangles.

#### Solution Preview

• Find an estimate of the area under the graph of between and above the -axis. Use four left endpoint rectangles.

The curve y = x2 is shown in diagram. At x = 0; y = 02 = 0, hence the curve is passing through the origin (0, 0) and at x = 2; y = 22 = 4 hence the curve is passing through the point (2, 4).

As to estimate the area under the curve from x = 0 to x = 2 we have to use four rectangles, the width of each rectangle should be (2 - 0)/4 = ½ = 0.5.

Now as we have to take left endpoint rectangles and the left points of the rectangles will have x coordinates

x = 0, 0.5, 1.0 and 1.5 respectively and their heights will be
y = 0, 0.25, 1.0 and 2.25.

The height of the first rectangle is zero and hence its area is 0.5*0 = 0

The height of the second rectangle is 0.25 and hence its area is 0.5*0.25 = 0.125

The height of the third rectangle is 1 and hence its area is 0.5*1 = 0.5 and

The height of the fourth rectangle is 2.25 and ...

#### Solution Summary

Area under the curve for a given range is calculated in different ways.

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