Finding the Area Bound by a Function and the x-axis

To find the area bound by the function y = 2x - 3 and the x-axis for x = 2 to x = 4, we can use integration. Integration allows us to calculate the area under a curve by finding the definite integral of the function within a given interval.

Step 1: Understanding the Given Function
The given function is y = 2x - 3. This is a linear function with a slope of 2 and a y-intercept of -3. We need to find the area bound by the function and the x-axis within the interval x = 2 to x = 4.

Step 2: Graphing the Function
To visualize the function and the area we are looking for, let's graph the function y = 2x - 3 within the given interval.

- Plot the points (2, 1) and (4, 5) on a coordinate plane.
- Draw a straight line passing through these two points. This line represents the function y = 2x - 3.

The graph should show a straight line rising from (2, 1) to (4, 5).

Step 3: Finding the Area
To find the area bound by the function and the x-axis within the interval x = 2 to x = 4, we need to calculate the definite integral of the function y = 2x - 3 within this interval.

The definite integral of a function f(x) from a to b is denoted as ∫[a to b] f(x) dx.

In this case, we want to find ∫[2 to 4] (2x - 3) dx.

To evaluate this integral, we can use the power rule of integration. The power rule states that the integral of x^n dx is (x^(n+1))/(n+1).

Applying the power rule to the given function, we have:
∫[2 to 4] (2x - 3) dx = [x^2 - 3x] evaluated from 2 to 4.

Now, let's evaluate the definite integral:
∫[2 to 4] (2x - 3) dx = [4^2 - 3(4)] - [2^2 - 3(2)].

Simplifying further, we get:
∫[2 to 4] (2x - 3) dx = [16 - 12] - [4 - 6] = 4.

Step 4: Interpreting the Result
The result of the definite integral is the area bound by the function y = 2x - 3 and the x-axis within the interval x = 2 to x = 4. In this case, the area is 4 square units.

Therefore, the area bound by the function y = 2x - 3 and the x-axis for x = 2 to x = 4 is 4 square units.

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