A left Riemann sum, a right Riemann sum, and a trapezoidal sum are used to approximate the value of \(\int\limits_0^1 f(x)   dx \), each using the same number of subintervals. The graph of the function \(f\) is shown in the figure above. Which of the sums give an overestimate of the value of \(\int\limits_0^1 f(x)   dx \) ?

• Left sum
• Right sum
• Trapezoidal Sum

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The sketches below give us an idea of whether each sum is an overestimate or underestimate. For an increasing and concave down function, only the right Riemann sum is an overestimate.

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Left sum:

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Right sum:

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Trapezoidal sum: