Riemann Sum Calculator for a Function

Approximate an integral (given by a function) using the Riemann sum step by step

The calculator will approximate the definite integral using the Riemann sum and the sample points of your choice: left endpoints, right endpoints, midpoints, or trapezoids.

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Your Input

Approximate the integral 02x4+13dx\int\limits_{0}^{2} \sqrt[3]{x^{4} + 1}\, dx with n=4n = 4 using the left Riemann sum.

Solution

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

abf(x)dxΔx(f(x0)+f(x1)+f(x2)++f(xn2)+f(xn1))\int\limits_{a}^{b} f{\left(x \right)}\, dx\approx \Delta x \left(f{\left(x_{0} \right)} + f{\left(x_{1} \right)} + f{\left(x_{2} \right)}+\dots+f{\left(x_{n-2} \right)} + f{\left(x_{n-1} \right)}\right)

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

We have that f(x)=x4+13f{\left(x \right)} = \sqrt[3]{x^{4} + 1}, a=0a = 0, b=2b = 2, and n=4n = 4.

Therefore, Δx=204=12\Delta x = \frac{2 - 0}{4} = \frac{1}{2}.

Divide the interval [0,2]\left[0, 2\right] into n=4n = 4 subintervals of the length Δx=12\Delta x = \frac{1}{2} with the following endpoints: a=0a = 0, 12\frac{1}{2}, 11, 32\frac{3}{2}, 2=b2 = b.

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

f(x0)=f(0)=1f{\left(x_{0} \right)} = f{\left(0 \right)} = 1

f(x1)=f(12)=17322341.020413775479337f{\left(x_{1} \right)} = f{\left(\frac{1}{2} \right)} = \frac{\sqrt[3]{17} \cdot 2^{\frac{2}{3}}}{4}\approx 1.020413775479337

f(x2)=f(1)=231.259921049894873f{\left(x_{2} \right)} = f{\left(1 \right)} = \sqrt[3]{2}\approx 1.259921049894873

f(x3)=f(32)=22397341.82340825744217f{\left(x_{3} \right)} = f{\left(\frac{3}{2} \right)} = \frac{2^{\frac{2}{3}} \sqrt[3]{97}}{4}\approx 1.82340825744217

Finally, just sum up the above values and multiply by Δx=12\Delta x = \frac{1}{2}: 12(1+1.020413775479337+1.259921049894873+1.82340825744217)=2.55187154140819.\frac{1}{2} \left(1 + 1.020413775479337 + 1.259921049894873 + 1.82340825744217\right) = 2.55187154140819.

Answer

02x4+13dx2.55187154140819\int\limits_{0}^{2} \sqrt[3]{x^{4} + 1}\, dx\approx 2.55187154140819A