# How do you evaluate the definite integral by the limit definition given #int (2x+5)dx# from [0,2]?

Here is a limit definition of the definite integral. (I hope it's the one you are using.)

Let's do one small step at a time.

Evaluate the sums

(We used summation formulas for the sums in the previous step.)

Rewrite before finding the limit

To finish the calculation, we have

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To evaluate the definite integral (\int_0^2 (2x+5)dx) using the limit definition, follow these steps:

- Start with the integral expression: (\int_a^b f(x)dx).
- Split the interval ([a, b]) into (n) subintervals of equal width: (\Delta x = \frac{b-a}{n}).
- Choose (x_i^
*) as the representative point in the (i)th subinterval, where (x_i^*) lies in the interval ([x_{i-1}, x_i]). - Form the Riemann sum: (R_n = \sum_{i=1}^n f(x_i^*)\Delta x).
- Take the limit of (R_n) as (n) approaches infinity: (\lim_{n\to\infty} R_n).
- This limit represents the value of the definite integral: (\int_a^b f(x)dx).

For the given integral (\int_0^2 (2x+5)dx):

- (a = 0) and (b = 2).
- (f(x) = 2x+5).
- Calculate (\Delta x = \frac{2-0}{n} = \frac{2}{n}).
- Choose (x_i^* = \frac{2i}{n}).
- Form the Riemann sum: [R_n = \sum_{i=1}^n f(x_i^*)\Delta x] [= \sum_{i=1}^n \left(2\left(\frac{2i}{n}\right) + 5\right)\frac{2}{n}] [= \frac{4}{n} \sum_{i=1}^n i + \frac{10}{n} \sum_{i=1}^n 1] [= \frac{4}{n} \cdot \frac{n(n+1)}{2} + \frac{10}{n} \cdot n] [= 2(n+1) + 10] [= 2n + 2 + 10] [= 2n + 12].
- Take the limit: [\lim_{n\to\infty} R_n = \lim_{n\to\infty} (2n + 12) = \infty.]

Therefore, the value of the definite integral (\int_0^2 (2x+5)dx) is infinite.

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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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