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Section 5.5 : Area Problem
For problems 1 – 3 estimate the area of the region between the function and the x-axis on the given interval using \(n = 6\) and using,
- the right end points of the subintervals for the height of the rectangles,
- the left end points of the subintervals for the height of the rectangles and,
- the midpoints of the subintervals for the height of the rectangles.
- \(f\left( x \right) = {x^3} - 2{x^2} + 4\) on \(\left[ {1,4} \right]\) Solution
- \(g\left( x \right) = 4 - \sqrt {{x^2} + 2} \) on \(\left[ { - 1,3} \right]\) Solution
- \(\displaystyle h\left( x \right) = - x\cos \left( {\frac{x}{3}} \right)\) on \(\left[ {0,3} \right]\) Solution
- Estimate the net area between \(f\left( x \right) = 8{x^2} - {x^5} - 12\) and the x-axis on \(\left[ { - 2,2} \right]\) 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? Solution