# More integral

### Newton-Cotes Algorithms for Equations
### Adaptive Quadrature


## Problems 

## Numerical Integration to Compute Work
In physics we've learned that work is computed
$$
Work = force * distance
$$
This can be written in int's integral form:
$$
W = \int{F(x)dx}
$$
If F(x) is easy to integrate, we could solve this problem analytically. However, a realistic problem the force may not be available to you as a function, but rather, tabulated data. Suppose some measurements were take of when a weighted box was pulled with a wire. If we data of the force on the wire and the angle of the wire from the horizontal plane.

| x (ft) | F(x) (lb) | θ (rad) | F(x) cos θ |
| ------ | --------- | ------- | ---------- |
| 0      | 0.0       | 0.50    | 0.0000     |
| 5      | 9.0       | 1.40    | 1.5297     |
| 10     | 13.0      | 0.75    | 9.5120     |
| 15     | 14.0      | 0.90    | 8.7025     |
| 20     | 10.5      | 1.30    | 2.8087     |
| 25     | 12.0      | 1.48    | 1.0881     |
| 30     | 5.0       | 1.50    | 0.3537     |
|        |           |         |            |
Use the trapezoidal rule to compute the work done on the box.

## Implementing the Composite Trapezoidal Rule

Implement a Python function to approximate integrals using the trapezoidal rule.

```python
import numpy as np

def trapz(f, a, b, n):
    x = np.linspace(a, b, n+1)
    y = f(x)
    h = (b - a) / n
    return h * (0.5*y[0] + y[1:-1].sum() + 0.5*y[-1])

# Example tests
f1 = np.sin
I_true1 = 2.0   # ∫_0^π sin(x) dx
for n in [4, 8, 16, 32]:
    print(n, trapz(f1, 0, np.pi, n))
```

Compare the results for increasing $n$ and observe how the error decreases with $O(h^2$).