# What is Integration Using the Trapezoidal Rule?

We can approximate the definite integral

by Trapezoid Rule

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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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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.

- How do you estimate the area under the graph of #f(x) = sqrt x# from #x=0# to #x=4# using four approximating rectangles and right endpoints?
- How do you use Riemann sums to evaluate the area under the curve of #f(x) = 2-x^2# on the closed interval [0,2], with n=4 rectangles using midpoint?
- If the area under the curve of f(x) = 25 – x2 from x = –4 to x = 0 is estimated using four approximating rectangles and left endpoints, will the estimate be an underestimate or overestimate?
- How do you use the trapezoidal rule to approximate the Integral from 0 to 0.5 of #(1-x^2)^0.5 dx# with n=4 intervals?
- How do you find the Riemann sum for this integral using right endpoints and n=3 for the integral #int (2x^2+2x+6)dx# with a = 5 and b = 11?

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