Right Riemann Sum to Determine an Integral that Defines an Area

Question

Consider the region bounded by y=2-x² and y=-x from x=-1 to x=2. Use a Right Riemann Sum to create an integral that calculates the area of the region. You don't need to evaluate the integral.

Answer 1

Δx = (2 - (-1))/n = 3/n

Subinterval right endpoints are x₁ = -1 + 3/n, x₂ = -1 + 2(3/n). x₃ = -1 + 3(3/n), ... x_n = -1 + n(3/n) = 2

Right Riemann Sum for n: ∑_{(I=1 to n)} [2 - (-1+ 3i/n)² + (-1 + 3i/n)] (3/n)

Area = limit as n approaches infinity of the sum above = ∫_{(-1 to 2)} [2 - x² + x] dx

Answer 2

delta= (b-a)/n = (2 - (-1))/n = 3/n

Integral = sigma (f2(xi) - f1(xi)) delta

xi = a+ i delta

i = 1 .. n

Integral = sigma (2- xi^2+xi) delta