# How do you approximate of #int sinx(dx)# from #[0,pi]# by the trapezoidal approximation using n=10?

See the explanation section.

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To approximate the integral of ( \sin(x) ) from ( 0 ) to ( \pi ) using the trapezoidal approximation with ( n = 10 ), follow these steps:

- Divide the interval ([0,\pi]) into ( n ) equal subintervals of width ( \Delta x = \frac{\pi}{n} ).
- Use the trapezoidal rule formula to calculate the area of each trapezoid formed by adjacent subintervals and the function values at their endpoints.
- Sum up the areas of all the trapezoids to approximate the total area.

The trapezoidal rule formula for each subinterval is:

[ A_i = \frac{1}{2} \left( f(x_i) + f(x_{i+1}) \right) \Delta x ]

where ( f(x_i) ) and ( f(x_{i+1}) ) are the function values at the endpoints of the ( i )th subinterval.

In this case, with ( f(x) = \sin(x) ), you substitute the values of ( x_i ) and ( x_{i+1} ) into the function, calculate the area of each trapezoid, and then sum up the areas of all trapezoids to approximate the integral.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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