# Two questions involving integration

• April 12th 2011, 06:41 PM
paulrb
Two questions involving integration
http://i.imgur.com/IGRB6.png

I cannot use integration by parts or numerical methods since that has not been introduced in the course. I know if I find a lower Riemann sum that equals the term on the left and an upper Riemann sum that equals the term on the right, that will work. But if that is the correct approach, then I'm not sure how to pick them.

edit: sorry just one question, I solved the other one but forgot to change the title :)

edit 2: actually what if you choose a partition P with two points {1, 2}.

Then 1/2e(e-1) < e < integral < e^2/2 < e(e-1)

since e is the lower Riemann sum for P and e^2/2 is the upper Riemann sum for P

is this correct?
• April 12th 2011, 07:12 PM
Prove It
Note that in the region $\displaystyle -1 \leq x \leq 2$,

$\displaystyle \frac{e^x}{2} \leq \frac{e^x}{x} \leq e^x$.

So $\displaystyle \int_1^2{\frac{e^x}{2}\,dx} \leq \int_1^2{\frac{e^x}{x}\,dx} \leq \int_1^2{e^x\,dx}$.