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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. the right end points of the subintervals for the height of the rectangles,
  2. the left end points of the subintervals for the height of the rectangles and,
  3. the midpoints of the subintervals for the height of the rectangles.

  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 \(\left[ { - 1,1} \right]\)