MAT136H5 Lecture 1: 5.2 The Definite Integral Explored

Graph the function  and evaluate the left and right endpoint sums provided that  and  . Identify if the area under the curve is in between the left and right endpoint sums.

Answer question parts a-d below. Give answers correct to the tenths.

(a) Use Riemann Sums to approximate area under the curve of f(x) = x² + 4 from 0 to 4 using 4 left rectangles.
(b) Use Riemann Sums to approximate area under the curve of f(x) = x² + 4 from 0 to 4 using 4 right rectangles.
(c) Using the sum of the left and right rectangles, find the average area.
(d) Find the value of the exact area using the integral.

In the following exercises, graph the function then use a calculator or a computer program to evaluate the following left and right endpoint sums. Is the area under the curve between the left and the right endpoint sums?