Write Integral as Series Calculator
Writing integrals as series is a powerful technique in calculus that allows us to approximate the value of a definite integral. It’s crucial for understanding and applying the fundamental theorem of calculus, as well as for numerical methods in calculus and other fields.
How to Use This Calculator
- Enter the integral you want to write as a series in the ‘Write the integral’ field.
- Enter the variable of integration in the ‘Variable of integration’ field.
- Enter the limit of integration in the ‘Limit of integration’ field.
- Click the ‘Calculate’ button to see the series and its approximation.
Formula & Methodology
The formula for writing an integral as a series is based on the definition of a definite integral and the binomial theorem. The series is given by:
∫abf(x) dx = (b – a) * f(a) + (b – a)2/2! * f'(a) + (b – a)3/3! * f”(a) + …
Real-World Examples
Example 1: Write ∫02x2 dx as a series.
Using the calculator, enter x2 as the integral, x as the variable, and 2 as the limit. The calculator will output the series and its approximation.
Example 2: Write ∫13ex dx as a series.
Enter ex as the integral, x as the variable, and 3 as the limit. The calculator will output the series and its approximation.
Example 3: Write ∫0πsin(x) dx as a series.
Enter sin(x) as the integral, x as the variable, and π as the limit. The calculator will output the series and its approximation.
Data & Statistics
| Integral | Exact Value | Approximate Value (n=10) |
|---|---|---|
| ∫02x2 dx | 1/3 | 0.3333 |
| ∫13ex dx | e3 – e | 20.086 |
| ∫0πsin(x) dx | cos(π) – cos(0) | -2.0000 |
Expert Tips
- Increasing the value of n in the series will generally increase the accuracy of the approximation.
- Be careful when using this method for integrals with singularities or discontinuities.
- This method can be used to approximate the value of definite integrals that cannot be evaluated exactly.
Interactive FAQ
What is the error bound for the series approximation?
The error bound for the series approximation is given by (b – a)n+1/(n+1)! * max{|f^(n+1)(x)| : x ∈ [a, b]}.
Can I use this method to approximate indefinite integrals?
No, this method is only valid for definite integrals.
Link to a detailed explanation of the formula and methodology
Link to another calculator for writing integrals as series
Link to a poster presentation on integrals as series