Question

find the lower sum for the region bounded by f(x) = 25 - x^2 and the x-axis between x=0 and x=5

Answer 1

Let y=f(x). The lower sum is that area above the x-axis. When x²-25=0, x=5 is the x-intercept on the right of the y-axis. This is the upper limit for the integration and 0 is the low limit:

Area =∫[0,5]ydx=∫[0,5](25-x²)dx=[25x-x³/3]₀⁵.

So area =(125-125/3)=250/3 square units (83.33).