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To solve the given problem, we need to evaluate the following expression:

\[ \sum_{i=1}^{100} \int_i^{i+2} x \, dx \]

1. First, let's find the indefinite integral of \(x\):

\[ \int x \, dx = \frac{x^2}{2} + C \]

2. Next, we evaluate the definite integral \(\int_i^{i+2} x \, dx\):

\[ \int_i^{i+2} x \, dx = \left[ \frac{x^2}{2} \right]_i^{i+2} = \frac{(i+2)^2}{2} - \frac{i^2}{2} \]

3. Simplify the expression:

\[ \frac{(i+2)^2}{2} - \frac{i^2}{2} = \frac{i^2 + 4i + 4 - i^2}{2} = \frac{4i + 4}{2} = 2i + 2 \]

4. Now, we sum this result from \(i = 1\) to \(i = 100\):

\[ \sum_{i=1}^{100} (2i + 2) \]

5. We can split this sum into two separate sums:

\[ \sum_{i=1}^{100} 2i + \sum_{i=1}^{100} 2 \]

6. Evaluate each sum separately. The first sum is an arithmetic series:

\[ \sum_{i=1}^{100} 2i = 2 \sum_{i=1}^{100} i = 2 \left( \frac{100 \cdot 101}{2} \right) = 100 \cdot 101 = 10100 \]

The second sum is a constant series:

\[ \sum_{i=1}^{100} 2 = 2 \cdot 100 = 200 \]

7. Add the results together:

\[ 10100 + 200 = 10300 \]

Therefore, the value of the given expression is:

\[ \boxed{10300} \]

Would you like any more details or have any other questions? Here are five questions you might want to ask:

1. How do you evaluate a definite integral?
2. Can you explain the properties of summation?
3. What is an arithmetic series, and how is it summed?
4. How do definite integrals differ from indefinite integrals?
5. Can you provide more examples of evaluating integrals with variable limits?

**Tip:** When dealing with integrals and summations, it's often useful to simplify the problem by breaking it into smaller, more manageable parts.

Learn how to evaluate the summation and definite integrals using step-by-step calculations from 1 to 100. This problem covers integral calculus and arithmetic series concepts, including the formula for the sum of an arithmetic series. Suitable for students in Grades 11-12.

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