Help with Riemann Sums and Definite Integrals
I have exams coming up, and there are some questions from my homework that I am still not sure of...please help me anyway you can, whether by showing me the steps or just giving me some help to start me off.
For the first 3, I'm not sure how to even start these:
1. (a) Find an upper bound for closed integral [1,6] for sqrt(x)*dx by calculating the upper Riemann sum, using the partition x0=1, x1=3, x2=6 of the interval [1,6].
(b) Find a lower bound for closed integral [1,6] for sqrt(x)*dx by calculating the lower Riemann sum, using the partition x0=1, x1=3, x2=6 of the interval [1,6]
(c) Use the Riemann sum corresponding to 5 inscribed rectangles of equal width to approximate the closed integral [1,3] (1/x)*dx.
3. Evaluate the closed integral [0,8] for (x*sqrt(1+x))*dx. I know that you would let u=(1+x), and then du=1*dx, but I'm not sure what my new integral with "u" would look like.
4. Evaluate the closed integral [-3,4] for abs(abs(x)-4)*dx
6. Evaluate closed integral [1,8] for ((4*(x^(2/3)+14)^3)/cuberoot(x))*dx
7. Calculate d/dt (closed integral [t^2, 2] for sqrt(x+1)*dx)
8. Find the average value of f(x)=(x^2+6x-5) on [1,4]