The Riemann Sum and the Definite Integral

 $\int_{-1}^22x\left(3-x\right)dx$  

 $\int_{-1}^1\left(x^3-5x\right)dx$  

 $\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}\left(\sin\ x\right)dx$  

 $\int_3^5\left(x\sqrt{x^2-1}\right)dx$  

 $\int_2^2\left(x^2-5x+2\right)dx$  

 Approximate the area bounded by  $f\left(x\right)=6x^2-3$  at the closed interval  $\left[-0.4,0.4\right]$  .

Using the Left Riemann Sum, approximate the area bounded by $y=\frac{1}{4}x^2+2$  from  $x=0$  to  $x=4$  when  $n=4$ .

Using the Right Riemann Sum, approximate the area bounded by $y=-\frac{1}{3}x^2+5$  from  $x=0$  to  $x=2$  when  $n=2$  .

Using the Midpoint Riemann Sum, approximate the area bounded by $y=x^2$  from  $x=0$  to  $x=2$  when  $n=4$  .

A doorway is parabolic in shape defined by $f\left(x\right)=-x^2+4$  . If an aluminum door will be installed in the said doorway and the base is 4 feet wide, at least how many square feet of aluminum will be used assuming that 1 unit is equal to 1 foot