Mathematics

Mathematics

Problem

Using a right Riemann sum with 2 equal subintervals, an approximation to ∫02x2 dx\displaystyle\int_{0}^{2}x^{2}\,dx is calculated. What is the percent error of this approximation, to the nearest tenth of a percent? a) 46.7\% b) 62.5\% c) 87.5\% d) 12.5\%
Intuitive explanation

The key is understanding that a right Riemann sum uses the right endpoint of each subinterval to build rectangles—which will overestimate this increasing function. You'll calculate the sum, compare it to the exact integral value, then find the percent error using ∣approx−exact∣exact×100%\frac{|\text{approx} - \text{exact}|}{\text{exact}} \times 100\%. The challenge is being careful with your subinterval widths and endpoint values.

Intuitive explanation

The key is understanding that a right Riemann sum uses the right endpoint of each subinterval to build rectangles—which will overestimate this increasing function. You'll calculate the sum, compare it to the exact integral value, then find the percent error using ∣approx−exact∣exact×100%\frac{|\text{approx} - \text{exact}|}{\text{exact}} \times 100\%. The challenge is being careful with your subinterval widths and endpoint values.

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