What is Integration Using the Trapezoidal Rule?

Answer 1
Let us divide the interval #[a,b]# into n subintervals of equal lengths.
#[a,b] to {[x_0,x_1], [x_1,x_2],[x_2,x_3],...,[x_{n-1},x_n]}#,
where #a=x_0 < x_1 < x_2 < cdots < x_n=b#.

We can approximate the definite integral

#int_a^b f(x)dx#

by Trapezoid Rule

#T_n=[f(x_0)+2f(x_1)+2f(x_2)+cdots2f(x_{n-1})+f(x_n)]{b-a}/{2n}#
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Answer 2

Integration using the Trapezoidal Rule is a numerical method for approximating the definite integral of a function. It divides the area under the curve into trapezoids and sums up their areas to estimate the integral. The formula for the Trapezoidal Rule is:

[ \int_{a}^{b} f(x) ,dx \approx \frac{b-a}{2n} [f(a) + 2f(x_1) + 2f(x_2) + \ldots + 2f(x_{n-1}) + f(b)] ]

where (n) is the number of subintervals (trapezoids), (a) and (b) are the lower and upper limits of integration, and (x_i) are the points at which the function is evaluated within each subinterval. The Trapezoidal Rule provides an approximation to the integral, and typically, the more subintervals used (the larger (n)), the more accurate the approximation becomes.

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Answer from HIX Tutor

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.

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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