Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best viewed in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (you should be able to scroll/swipe to see them) and some of the menu items will be cut off due to the narrow screen width.
Assignment Problems Notice
Please do not email me to get solutions and/or answers to these problems. I will not give them out under any circumstances nor will I respond to any requests to do so. The intent of these problems is for instructors to use them for assignments and having solutions/answers easily available defeats that purpose.
If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
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,
- 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.
- \(f\left( x \right) = 15 + 4x - {x^3}\) on \(\left[ {1,3} \right]\)
- \(g\left( x \right) = - 3{x^2} + 2x - 1\) on \(\left[ { - 4,0} \right]\)
- \(h\left( x \right) = 8\ln \left( x \right) - x\) on \(\left[ {2,6} \right]\)
- \(f\left( x \right) = {\sin ^2}\left( {\frac{x}{2}} \right)\) on \(\left[ {0,3} \right]\)
- \(g\left( x \right) = \sin \left( x \right)\cos \left( x \right) - 1\) 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?
- \(h\left( x \right) = 8x - \sqrt {x + 4} \) on \(\left[ { - 3,2} \right]\)
- \(g\left( x \right) = 5 + x - {x^2}\) on \(\left[ {0,4} \right]\)
- \(f\left( x \right) = x{{\bf{e}}^{ - {x^{\,2}}}}\) on \(\left[ { - 1,1} \right]\)