Lower sum

Definition

A lower sum is an approximation of the area under a curve using the sum of the areas of rectangles that lie entirely below the curve. It is calculated by taking the infimum (or minimum) value of the function within each subinterval.

5 Must Know Facts For Your Next Test

1. Lower sums provide an underestimate of the actual area under a curve.
2. Each rectangle in a lower sum calculation has a height determined by the minimum function value in its subinterval.
3. As the number of subintervals increases, the approximation given by the lower sum becomes more accurate.
4. The concept of lower sums is essential for understanding Riemann sums and definite integrals.
5. Lower sums are often used in conjunction with upper sums to bound the true value of an integral.

Review Questions

• How do you determine the height of each rectangle when calculating a lower sum?
• Why do lower sums generally underestimate the actual area under a curve?
• What happens to the accuracy of a lower sum as you increase the number of subintervals?

Related terms

Upper Sum: An approximation of the area under a curve using rectangles that lie above or touch the curve, calculated using supremum values within each subinterval.

Riemann Sum: A method for approximating integrals by dividing an interval into smaller subintervals and summing up areas calculated at specific points within those intervals.

Definite Integral: \( \int_a^b f(x) \, dx \) represents the exact area under a function $f(x)$ from $a$ to $b$, which can be approximated using Riemann sums.