Calculus II

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Lower sum

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Calculus II

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.

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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

  • 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.

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