91Ó°ÊÓ

A definite integral is a fundamental concept in calculus. It allows us to find the exact area under a curve between two points. When calculating definite integrals, we add up tiny rectangles under the curve in a specific interval. The formula for the definite integral of a function \( f(x) \) from \( a \) to \( b \) is written as:
\[ \int_{a}^{b} f(x) \, dx \]
The definite integral has several properties:

• It gives the net area between the curve and the x-axis.
• Areas above the x-axis are positive, while those below are negative.
• We can use it to find areas between two curves by integrating the difference of their functions.

The problem in your exercise uses these properties to find the area between the curves \( y = e^x \) and \( y = \frac{1}{x^2} \) from \( x = 1 \) to \( x = 2 \). To set up the integral, we take the difference of these two functions: \( \int_{1}^{2} \big(e^x - \frac{1}{x^2}\big) \,dx \). This represents the area between these curves in the given interval.

Integration is another core concept in calculus, closely related to differentiation. While differentiation finds the rate of change, integration finds the total accumulation. Integration can be thought of as the reverse process of differentiation. There are different methods of integration, such as:

• Indefinite Integration: Finding a general form of the antiderivative of a function.
• Definite Integration: Calculating the exact area under a curve between two points.

For your problem, you used definite integration to find the area between the curves. You first integrated the function \( e^x \) on its own:
\[ \int e^x \, dx = e^x \]
Then, you integrated the function \( -\frac{1}{x^2} \):
\[-\int \frac{1}{x^2} \, dx = \int -x^{-2} \, dx = \frac{1}{x} \]
Evaluating these from \( x = 1 \) to \( x = 2 \) gives you the two areas needed to find the total area between the curves.

An exponential function is a mathematical function of the form \( f(x) = a^x \), where \( a \) is a positive constant. A commonly used base is the natural exponential function \( e^x \), where \( e \) (approximately 2.718) is a unique number called Euler's number. Exponential functions have distinct properties:

• They grow rapidly as x increases if \( a > 1 \).
• Their derivative \( f'(x) \) is proportional to the function itself \( f'(x) = a^x \, \ln(a) \).
• The integral of \( e^x \) is also \( e^x \).

In your exercise, understanding exponential functions helps you integrate \( e^x \) quickly. Specifically, you used:
\[ \int e^x \, dx = e^x \]
This simple property allowed you to focus more on the process of adding and subtracting areas within the definite interval to find the area between the curves.