© 2016 Shmoop University, Inc. All rights reserved.
Definite Integrals

Definite Integrals

Left-Hand Sum Examples

Example 1

Let R be the region between the graph y = f (x) = x2 + 1 and the x-axis on the interval [0, 4]:

Use a left-hand sum with two sub-intervals to approximate the area of R.

Example 2

Let R be the region between the graph y = f (x) = x2+ 1 and the x-axis on the interval [0, 4]. Use a Left-Hand Sum with 4 sub-intervals to estimate the area of R.

Example 3

Let S be the region between the graph of g and the x-axis on the interval [0, 4].

Use a left-hand sum with 2 sub-intervals to estimate the area of S. Is this an under-estimate or an over-estimate?

Example 4

  • Let W be the region between the graph of f and the x-axis on the interval [-4, 0].

  • Use a left-hand sum with 2 sub-intervals to estimate the area of Z.
  • Use a left-hand sum with 4 sub-intervals to estimate the area of Z.

Example 5

Let f (x) = x2 + 2 and let R be the region between the graph of f and the x-axis on the interval [0, 8]. Use a left-hand sum with 4 sub-intervals to estimate the area of R.

Example 6

Let f(x) = 4x and let R be the region between the graph of f and the x-axis on the interval [1, 2]. Use a left-hand sum with 4 sub-intervals to estimate the area of R.

Example 7

Let f (x) = 2x on [2,10]. Find LHS(5). That is, use a left-hand sum with 5 sub-intervals to estimate the area between the graph of f and the x-axis on [2, 10].

Example 8

Use a left-hand sum to estimate the area between the graph of g and the x-axis on the interval [0,10].