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4-45.

Approximate $A(f,1\le x\le2)$ for $f(x)=-x^2+6$.

1. Use four rectangles to approximate an underestimate of the area under the curve.

Sketch the curve and divide up the interval into four equal sections using vertical lines. Draw in rectangles that are below the curve.

2. Use four rectangles to approximate an overestimate of the area under the curve.

See the hint in part (a). This time the tops of the rectangles should be above the curve.

3. Express the sum of the underestimate using sigma notation.

$0.25 \displaystyle\sum_{k=1}^{4}-(\text{find this expression})^{2}+6$