Question

Consider the function f(x)=7x+8x^2 over the interval [0,1].

1) Write down a Riemann sum for f(x) over the given interval which is guaranteed to be an overestimate.
2) Using the formulas ∑k=1nk=n(n+1)2
and ∑k=1nk2=n(n+1)(2n+1)6 , write down the above Riemann sum without using a Σ

Answer 1

To find an overestimate Riemann sum for the given function over the interval [0,1], divide the interval into n subintervals of equal width and take the right endpoints of each subinterval as sample points. The Riemann sum as an overestimate is (n + 1)/2.

Step-by-step explanation:

To write down a Riemann sum for the given function over the interval [0,1] that is guaranteed to be an overestimate, we need to use the right endpoints of each subinterval.

Let's start with dividing the interval [0,1] into subintervals of equal width. The width of each subinterval is Δ = (1−0)/ = 1/.

For each subinterval, we take the right endpoint as the sample point. So, the Riemann sum for the given function over the interval [0,1] as an overestimate is:

= ( + 1)/2 × Δ

= ( + 1)/2 × (1/)

= ( + 1)/2.