91Ó°ÊÓ

In calculus, understanding the area under a curve is fundamental. When we talk about the area under a curve, it means the total region enclosed between the curve of the function and the x-axis over a specific interval. For the problem at hand, the curve described by the equation \( y = 5x - x^2 \) is a parabola that opens downwards, and we are interested in the portion of the area between \( x = 1 \) and \( x = 3 \).

• Imagine slicing this area vertically into thin rectangles or slices starting at \( x = 1 \) and ending at \( x = 3 \).
• Each rectangle's height is determined by the value of the function at that x-coordinate, e.g., \( y = 5x - x^2 \).
• The width of each slice is an infinitesimal value \( dx \), representing a tiny change in x.

The strength of this approach lies in its ability to sum these tiny areas together to approximate the entire region under the curve. This concept is a stepping stone to the integration process, where the sum of the tiny areas becomes a definite integral that gives us the exact area.

The Fundamental Theorem of Calculus is central to solving problems involving the area under a curve. This theorem allows us to find the exact area under a curve by connecting differentiation and integration—two core concepts of calculus. According to the theorem, if you have a continuous function defined on an interval, finding its antiderivative allows you to quickly compute its integral over that interval.

• First, identify the antiderivative (also known as the indefinite integral) of the given function, which in our case is \( 5x - x^2 \).
• Once you have the antiderivative, evaluate it at the bounds of the interval (here, from \( x = 1 \) to \( x = 3 \)).
• Subtract the value of the antiderivative at the lower bound from its value at the upper bound to find the area.

Using this method for the parabola \( y = 5x - x^2 \) over the interval from 1 to 3, we compute the area as \[ \int_{1}^{3} (5x - x^2) \, dx = \frac{9}{2} \]. This theorem not only simplifies the integration process but ensures precise calculations.

An antiderivative is essentially the inverse of differentiation, and it plays a significant role in finding the area under a curve through integration. Given a function, its antiderivative is another function whose derivative returns the original function. In our exercise, the function is \( f(x) = 5x - x^2 \). To find the antiderivative, we apply basic rules of integration:

• For \( 5x \), the antiderivative is \( \frac{5}{2}x^2 \).
• For \( -x^2 \), the antiderivative is \( - \frac{x^3}{3} \).

Therefore, the antiderivative of \( 5x - x^2 \) is \( \frac{5}{2}x^2 - \frac{x^3}{3} \). This step is crucial in the integration process because once we have the antiderivative, we can apply the Fundamental Theorem of Calculus to find the area under the curve. Evaluating this antiderivative at the given bounds (1 and 3) provides the precise area of the region. The concept of antiderivatives simplifies the whole process of integration by transforming it into a task of finding a function, evaluating it, and performing simple subtraction.