Right Endpoint Approximation Calculator for a Function

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

The right Riemann sum (also known as the right endpoint approximation) uses the right endpoint of a subinterval for computing the height of the approximating rectangle:

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

where $\Delta x = \frac{b - a}{n}$.

We have that $f{\left(x \right)} = \sqrt{\sin^{5}{\left(x \right)} + 1}$, $a = 1$, $b = 5$, and $n = 4$.

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.

$f{\left(x_{1} \right)} = f{\left(2 \right)} = \sqrt{\sin^{5}{\left(2 \right)} + 1}\approx 1.273431158532973$

$f{\left(x_{2} \right)} = f{\left(3 \right)} = \sqrt{\sin^{5}{\left(3 \right)} + 1}\approx 1.000027983813047$

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

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

Finally, just sum up the above values and multiply by $\Delta x = 1$: $1 \left(1.273431158532973 + 1.000027983813047 + 0.867027424870839 + 0.434954473370867\right) = 3.575441040587726.$

$\int\limits_{1}^{5} \sqrt{\sin^{5}{\left(x \right)} + 1}\, dx\approx 3.575441040587726$A