Introduction

In the previous post, we learned about the concept of area under curves and its significance in real-world applications. Now, let's dive deeper and focus on finding the area under linear functions.

Method

To find the area under a linear function in a specific interval, we can use definite integrals. The definite integral represents the signed area between the curve and the x-axis within the given interval.

Consider a linear function in the form of y = mx + c, where m represents the slope and c represents the y-intercept. To find the area under the linear function, we need to integrate the function over the desired interval.

Example

Let's find the area under the linear function y = 2x + 3 between x = 0 and x = 5.

∫[0,5] (2x + 3) dx

To solve this integral, we can use the power rule of integration:

[(x^2)/2 + 3x] [0,5]

Substituting the upper and lower limits of integration:

[(5^2)/2 + 3(5)] - [(0^2)/2 + 3(0)]
=(25/2 + 15) - (0/2 + 0)
=(25/2 + 30/2)
=55/2

Hence, the area under the linear function y = 2x + 3 between x = 0 and x = 5 is 55/2 square units.

Conclusion

Finding the area under linear functions using definite integrals allows us to calculate the accumulated value of the function within a given interval. It is a powerful tool that finds applications in various fields of mathematics and real-world problem-solving.

Keep practicing and exploring the exciting world of area under curves! You're doing great!