91Ó°ÊÓ

A definite integral helps us find the area under a curve within a given interval on the x-axis. It essentially sums up all the infinitesimal slices under the curve from one point to another, giving us a total area. In our exercise, we are finding the area under the curve \( \frac{1}{\sqrt{x}} \) from \( x = 1 \) to \( x = b \). This is set up by the integral \( \int_{1}^{b} \frac{1}{\sqrt{x}} \, dx = 6 \).

• The number "6" represents the total area under the curve between the points 1 and \( b \).
• The process involves evaluating the integral with specific limits.

Using the definite integral, we apply what is known as the Fundamental Theorem of Calculus. This theorem links the process of differentiation and integration, allowing us to evaluate the integral at specific bounds to find a precise numerical value. As demonstrated, this resulted in \( b \) needing to be 16 for the area to equal 6.

The concept of the area under a curve is fundamental in understanding real-world phenomena, such as physics, engineering, or economics. In essence, it quantifies the space that lies beneath a given curve and above the x-axis between two points.

• In our example, we look at the curve \( \frac{1}{\sqrt{x}} \).
• We calculate the area from \( x = 1 \) to some unknown \( x = b \).

The area under a curve can provide insights into various accumulated quantities. For example, it may represent distance traveled over a time period if the curve represented speed. By converting the function \( \frac{1}{\sqrt{x}} \) into its integrable form, \( x^{-1/2} \), we can calculate the enclosed area using the integral method, ultimately providing the solution \( b = 16 \).

The power rule of integration is a straightforward method to integrate functions of the form \( x^n \). When you have a function \( x^n \), where \( n \) is any real number except -1, the integral is given by:\[\int x^n \, dx = \frac{x^{n+1}}{n+1} + C.\]In our exercise, we needed to integrate \( \frac{1}{\sqrt{x}} \), which can be rewritten as \( x^{-1/2} \).

• Using the power rule, the integral becomes \( 2x^{1/2} + C \).
• This rule greatly simplifies finding antiderivatives for polynomial functions.

Once integrated, we use the result to find the specific values from the definite integral's limits, ultimately determining the specific value of \( b \) necessary to satisfy the given condition. Mastering the power rule is key to solving integrals quickly and efficiently.