Lower sum

Definition

The lower sum is an approximation of the area under a curve using the sum of the areas of inscribed rectangles. Each rectangle's height is determined by the minimum function value within each subinterval.

5 Must Know Facts For Your Next Test

1. The lower sum provides an underestimate of the actual area under a curve for non-negative functions.
2. It is calculated by summing up areas of rectangles whose heights are the minimum values in each partition subinterval.
3. $$L(f, P) = \sum_{i=1}^{n} m_i \Delta x_i$$ where $$m_i$$ is the minimum value of $$f(x)$$ on the i-th subinterval and $$\Delta x_i$$ is the width of that subinterval.
4. The accuracy of a lower sum improves as the number of subintervals increases.
5. Lower sums are often used in conjunction with upper sums to bound the true value of an integral.

Review Questions

Related terms

Upper Sum:

An approximation method for finding areas under curves using rectangles where heights are determined by maximum function values within each subinterval.

Riemann Sum:

A general method for approximating integrals by summing up areas of rectangles; can be left, right, midpoint, or trapezoidal sums.

Partition:

A division of an interval into smaller subintervals used for calculating Riemann sums and other approximations.