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.
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The lower sum provides an underestimate of the actual area under a curve for non-negative functions.
It is calculated by summing up areas of rectangles whose heights are the minimum values in each partition subinterval.
$$L(f, P) = \sum_{i=1}^{n} m_i \Delta x_i$$ 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.
The accuracy of a lower sum improves as the number of subintervals increases.
Lower sums are often used in conjunction with upper sums to bound the true value of an integral.
Review Questions
What distinguishes a lower sum from an upper sum?
How does increasing the number of subintervals affect the accuracy of a lower sum?
Write out the formula for calculating a lower sum.
An approximation method for finding areas under curves using rectangles where heights are determined by maximum function values within each subinterval.