Show All Notes Hide All Notes

Section 5.5 : Area Problem

For problems 1 – 5 estimate the area of the region between the function and the x-axis on the given interval using \(n = 6\) and using,

1. \(f\left( x \right) = 15 + 4x - {x^3}\) on \(\left[ {1,3} \right]\)
2. \(g\left( x \right) = - 3{x^2} + 2x - 1\) on \(\left[ { - 4,0} \right]\)
3. \(h\left( x \right) = 8\ln \left( x \right) - x\) on \(\left[ {2,6} \right]\)
4. \(f\left( x \right) = {\sin ^2}\left( {\frac{x}{2}} \right)\) on \(\left[ {0,3} \right]\)
5. \(g\left( x \right) = \sin \left( x \right)\cos \left( x \right) - 1\) on \(\left[ { - 2,1} \right]\)

For problems 6 – 8 estimate the net area between the function and the x-axis on the given interval using \(n = 8\) and the midpoints of the subintervals for the height of the rectangles. Without looking at a graph of the function on the interval does it appear that more of the area is above or below the x-axis?

1. \(h\left( x \right) = 8x - \sqrt {x + 4} \) on \(\left[ { - 3,2} \right]\)
2. \(g\left( x \right) = 5 + x - {x^2}\) on \(\left[ {0,4} \right]\)
3. \(f\left( x \right) = x{{\bf{e}}^{ - {x^{\,2}}}}\) on \(\left[ { - 1,1} \right]\)