Mastering Definite Integrals: Understanding the Concept and Solving for Area Under Curves

Definite Integrals

Definite integrals are a type of integral that allow us to find the exact area under a curve over a specific interval

Definite integrals are a type of integral that allow us to find the exact area under a curve over a specific interval. They are used in various mathematical and scientific fields to solve problems related to area, volume, and accumulation.

To understand definite integrals, we need to first understand the concept of integrals. An integral represents the area under a curve, and it is denoted by ∫ (sigma) symbol followed by the function to be integrated, such as ∫ f(x) dx. The integral sign represents the sum of infinitely many tiny areas (infinitesimal rectangles) that make up the total area under the curve.

Now, a definite integral is used when we want to find the area under the curve between two specific points, referred to as the lower limit (a) and the upper limit (b). The notation for a definite integral is ∫[a, b] f(x) dx.

To find the definite integral of a function, we follow these steps:

1. Evaluate the antiderivative of the function f(x), denoted as F(x), which is the same as finding the indefinite integral ∫ f(x) dx.
2. Substitute the upper limit (b) into the antiderivative F(x) and subtract the value obtained by substituting the lower limit (a) into F(x).
3. Write the definite integral as ∫[a, b] f(x) dx = F(b) – F(a).

Let’s consider an example to illustrate the process:

Example:
Find the definite integral ∫[0, 2] 3x^2 dx.

1. To find the antiderivative of 3x^2, we add 1 to the exponent and divide by the new exponent to get: F(x) = x^3.
2. Substitute the upper limit (2) into F(x): F(2) = 2^3 = 8.
3. Substitute the lower limit (0) into F(x): F(0) = 0^3 = 0.
4. Subtract the value from step 3 from the value from step 2: F(2) – F(0) = 8 – 0 = 8.

Therefore, the definite integral ∫[0, 2] 3x^2 dx equals 8. This means that the area under the curve of the function 3x^2 between x = 0 and x = 2 is equal to 8 square units.

Remember that definite integrals can also be negative if the curve lies below the x-axis within the given interval. In such cases, the area is considered negative.

I hope this explanation helps you understand definite integrals better. Don’t hesitate to ask if you have any further questions!

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