In calculus, an integral is the opposite of a derivative. It helps us calculate the total accumulation or area under a curve. Integrals are used to find quantities like the total distance traveled, the area of irregular shapes, or the accumulated change over time.

Key Points:

• Integral: The integral of a function gives the area under the curve of the function between two points. It is often used to calculate the total of something that accumulates over time, such as distance, area, or volume.
• Notation: The integral of a function f(x) from a to b is written as ∫_a^b f(x) dx. This represents the area under the curve of f(x) from x = a to x = b.
• Indefinite vs Definite Integrals:
  • Indefinite Integral: The result of an integral without specific limits. It represents a family of functions and is written as ∫ f(x) dx.
  • Definite Integral: The integral with specific limits, which gives the area under the curve between those limits. It is written as ∫_a^b f(x) dx.

Example 1: Basic Indefinite Integral

Let's take the function f(x) = x². To find its indefinite integral, we use the power rule for integration. The power rule says that if f(x) = x^n, then the integral of f(x) is (x^(n+1)) / (n+1).
Applying this to f(x) = x², we get:
∫ x² dx = (x³) / 3 + C, where C is the constant of integration.
This means that the total accumulation of x² is (x³) / 3 + C.

Example 2: Basic Definite Integral

Now, let's calculate the definite integral of f(x) = x² from 0 to 2.
Using the result from the previous example, the definite integral is:
∫_0^2 x² dx = [(x³) / 3] from 0 to 2.
Substitute the limits of integration:
= (2³ / 3) - (0³ / 3) = 8 / 3 - 0 = 8 / 3.
This means the area under the curve of f(x) = x² between x = 0 and x = 2 is 8/3.

Key Takeaways:

• An integral represents the total accumulation, such as area under a curve or total distance traveled.
• The process of integration is the reverse of differentiation, just as subtraction is the reverse of addition.
• Integrals can be either indefinite (general form) or definite (with specific limits). Definite integrals calculate the total quantity between two values.

Summary:

Integrals help us find the total accumulated value of a function over a specific range. Whether it’s calculating area under a curve, the total distance traveled, or the accumulated change, integrals are an essential tool in calculus. The process of integrating is like summing up small quantities to find the total result.

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