Approximating an E&M Integral with Calculus

• Matterwave
In summary, the conversation discusses a calculus problem that involves evaluating integrals with the use of constants Hc and H0. The second integral can be eliminated and the first can be computed using the fact that Hmax>>Hc, H0 to arrive at an approximate value.
Matterwave
Gold Member

Homework Statement

This is technically an E&M question, but I've reduced it to a calculus problem. Basically I have to evaluate:

$$B_0(\int_{-H_{max}}^{H_{max}}{tanh(\frac{H+H_c}{H_0})dH - \int_{-H_{max}}^{H_{max}}{tanh(\frac{H-H_c}{H_0})dH)$$

Where $$H_{max}>>H_C, H_0$$.

The Attempt at a Solution

I'm looking at this and I have no idea how to go about approximating this integral...I suppose I could just brute force the integrals and keep all the H_max and stuff, and then later see if i can approximate something...but the expressions are really quite long and I'd like to avoid that if I can. Is there a way?

Hi Matterwave!

Are Hc and H0 constants?

If so, that's just ∫tanh(Ax + b) dx …

and ∫tanh is ln(cosh)

You can eliminate the second integral (it's just the additive inverse of the first; prove it). Per tiny-tim's hint, you can compute the integral. Simplify and finally use the fact that Hmax>>Hc , H0 to arrive at an approximate value.

1. What is approximate integration?

Approximate integration is a method used in mathematics and science to estimate the value of a definite integral. It involves dividing the interval of integration into smaller subintervals and using the values of the function at certain points within each subinterval to approximate the area under the curve.

2. Why is approximate integration necessary?

Exact integration can be difficult or impossible to calculate in many cases, especially for complex functions. Approximate integration provides a close estimate of the value of a definite integral, making it a useful tool for solving problems in various fields.

3. What are the different methods of approximate integration?

There are several methods of approximate integration, including the Trapezoidal Rule, Simpson's Rule, and the Midpoint Rule. Each method uses a different approach to estimate the value of a definite integral and has its own strengths and limitations.

4. How accurate is approximate integration?

The accuracy of approximate integration depends on the method used and the number of subintervals used to approximate the integral. Generally, the more subintervals used, the more accurate the approximation will be. However, even with a large number of subintervals, there may still be some error in the approximation.

5. In what situations would approximate integration be useful?

Approximate integration is useful in many real-world scenarios, such as calculating the area under a curve in physics, determining the work done in a chemical reaction, or estimating the value of a financial investment. It can also be used when the exact value of a definite integral is unknown or too difficult to calculate.

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