Question

I've attached pictures, this is regarding Reimann's sum for integral calculus. SOS, both parts of the question please if possible. THANK YOU

Answer 1

a) Left-endpoint Riemann Sum = 0. (b) Right-endpoint Riemann Sum = 13.

Let's calculate the left-endpoint and right-endpoint Riemann sums for the given function f(x) over the interval [-4, 6], partitioned into subintervals [-4, -1], [-1, 0], [0, 1], [1, 3], [3, 5], [5, 6].

The function values at the left endpoints of each subinterval are as follows: f(-4) = -3, f(-1) = -1, f(0) = 0, f(1) = 1, f(3) = 2, f(5) = 4.

The function values at the right endpoints of each subinterval are: f(-1) = -1, f(0) = 0, f(1) = 1, f(3) = 2, f(5) = 4, f(6) = 3.

Now, let's calculate the left-endpoint and right-endpoint Riemann sums:

(a) Left-endpoint Riemann Sum: Riemann Sum = (-3 * 3) + (-1 * 1) + (0 * 1) + (1 * 2) + (2 * 2) + (4 * 1) = -9 - 1 + 0 + 2 + 4 + 4 = 0.

(b) Right-endpoint Riemann Sum: Riemann Sum = (-1 * 3) + (0 * 1) + (1 * 1) + (2 * 2) + (4 * 2) + (3 * 1) = -3 + 0 + 1 + 4 + 8 + 3 = 13.

Therefore, the left-endpoint Riemann sum is 0, and the right-endpoint Riemann sum is 13.