# Right Endpoint Approximation Calculator for a Function

An online calculator for approximating a definite integral using right endpoints (the right Riemann sum), with steps shown.

Related calculator: Right Endpoint Approximation Calculator for a Table

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Approximate the integral $$\int\limits_{1}^{5} \sqrt{\sin^{5}{\left(x \right)} + 1}\, dx$$$with $$n = 4$$$ using the right endpoint approximation.

## Solution

The right Riemann sum (also known as the right endpoint approximation) uses the right endpoints of a subinterval:

$$\int\limits_{a}^{b} f{\left(x \right)}\, dx\approx \Delta x \left(f{\left(x_{1} \right)} + f{\left(x_{2} \right)} + f{\left(x_{3} \right)}+\dots+f{\left(x_{n-1} \right)} + f{\left(x_{n} \right)}\right)$$$where $$\Delta x = \frac{b - a}{n}$$$.

We have that $$a = 1$$$, $$b = 5$$$, $$n = 4$$$. Therefore, $$\Delta x = \frac{5 - 1}{4} = 1$$$.

Divide the interval $$\left[1, 5\right]$$$into $$n = 4$$$ subintervals of the length $$\Delta x = 1$$$with the following endpoints: $$a = 1$$$, $$2$$$, $$3$$$, $$4$$$, $$5 = b$$$.

Now, just evaluate the function at the right endpoints of the subintervals.

$$f{\left(x_{1} \right)} = f{\left(2 \right)} = \sqrt{\sin^{5}{\left(2 \right)} + 1}\approx 1.27343115853297$$$$$f{\left(x_{2} \right)} = f{\left(3 \right)} = \sqrt{\sin^{5}{\left(3 \right)} + 1}\approx 1.00002798381305$$$

$$f{\left(x_{3} \right)} = f{\left(4 \right)} = \sqrt{\sin^{5}{\left(4 \right)} + 1}\approx 0.867027424870839$$$$$f{\left(x_{4} \right)} = f{\left(5 \right)} = \sqrt{\sin^{5}{\left(5 \right)} + 1}\approx 0.434954473370867$$$

Finally, just sum up the above values and multiply by $$\Delta x = 1$$$: $$1 \left(1.27343115853297 + 1.00002798381305 + 0.867027424870839 + 0.434954473370867\right) = 3.57544104058773.$$$

$$\int\limits_{1}^{5} \sqrt{\sin^{5}{\left(x \right)} + 1}\, dx\approx 3.57544104058773$$\$A