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Compute the right-endpoint Riemann sum

Consider the function f(x)=5−x310f(x) = 5 - \dfrac{x^3}{10} on the interval [0,4][0,4], divided into n=4n = 4 equal subintervals.

Partition [0,4][0,4] into n=4n = 4 subintervals of equal width and use the right endpoint of each subinterval to approximate the area under f(x)=5−x310f(x) = 5 - \dfrac{x^3}{10} on [0,4][0,4].

Right sum=∑i=14f(xi) Δx\text{Right sum} = \sum_{i=1}^{4} f(x_i)\,\Delta x

Compute the value of this right-endpoint Riemann sum.

Hint

First find Δx=(4−0)/4\Delta x = (4-0)/4, then list the right endpoints x1,x2,x3,x4x_1, x_2, x_3, x_4. Evaluate ff at each, add them up, and multiply the total by Δx\Delta x.

Step-by-step solution

The width of each subinterval is Δx=4−04=1\Delta x = \dfrac{4-0}{4} = 1. The right endpoints are x1=1x_1 = 1, x2=2x_2 = 2, x3=3x_3 = 3, x4=4x_4 = 4, since we start at a=0a=0 and step by Δx\Delta x each time, always landing on the right side of each subinterval.

Evaluate ff at each: f(1)=5−0.1=4.9f(1) = 5 - 0.1 = 4.9, f(2)=5−0.8=4.2f(2) = 5 - 0.8 = 4.2, f(3)=5−2.7=2.3f(3) = 5 - 2.7 = 2.3, f(4)=5−6.4=−1.4f(4) = 5 - 6.4 = -1.4. Summing these gives 4.9+4.2+2.3+(−1.4)=10.04.9+4.2+2.3+(-1.4) = 10.0, and multiplying by Δx=1\Delta x = 1 gives the right sum =10.0= 10.0.

Determine whether the right sum overestimates or underestimates the true area

Using the fact that f(x)=5−x310f(x) = 5 - \dfrac{x^3}{10} is decreasing on [0,4][0,4], determine whether the right-endpoint sum computed above is an overestimate or an underestimate of the exact area under the curve on [0,4][0,4]. State your conclusion and briefly justify it based on the behavior of ff.

Hint

Think about how the height used on each subinterval (the right endpoint value) compares to the actual heights of ff across that subinterval when ff is decreasing.

Step-by-step solution

Since f′(x)=−3x210≤0f'(x) = -\dfrac{3x^2}{10} \le 0 on [0,4][0,4], ff is decreasing throughout the interval. For a decreasing function, the right endpoint of each subinterval gives the smallest value of ff on that subinterval.

Because each rectangle's height (from the right endpoint) is less than or equal to the actual function values on that subinterval, the right sum underestimates the true area under the curve on [0,4][0,4].

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