Q1. A Riemann Sum should get closer and closer to the definite integral as the number n of subintervals gets bigger and bigger. Define LeftErr=abs(LeftSum-c). What is LeftErr when n=2, when n=5 and when n=10? Is it decreasing?
Q2. Define f(x)=3, a=1, b=5. Geometrically, what is the shape of the area under the curve in the interval [1,5]? Find this area A using a standard formula for area. What is the value of the definite integral ? Set n=1. What is the Left Riemann Sum? Why is it the same as the definite integral? Set n=2, then n=5 and then n=10. Why doesn’t the Left Riemann Sum change?
Q3. Define f(x)=-3 (a=1, b=5). Geometrically, what is the shape of the area under the curve in the interval [1,5]? Find this area A using a standard formula for area. What is the value of the definite integral ? Why is A different than I? Set n=1. What is the Left Riemann Sum? Why is it the same as the definite integral? Set n=2, then n=5 and then n=10. Why doesn’t the Left Riemann Sum change?
Q4. Define f(x)=sin(x), a=0, b=2*pi. What is the value of the definite integral ? What is LeftErr when n=2, when n=5 and when n=10? Why is the LeftSum so “accurate”?
Q5. Change b=pi/2. What is the value of the definite integral ? What is LeftErr when n=2, when n=5 and when n=10? Is LeftSum so “accurate” now?
