91Ó°ÊÓ

Definite integrals are used in calculus to calculate the accumulation of quantities. In this context, definite integrals help us find the area under a curve between two points. When we set up a definite integral, we need:

• the function defining the curve,
• the limits of integration, which represent the interval on the x-axis,
• and the variable of integration, often represented as \(dx\).

For this exercise, the function is \(y = \sqrt{x} - 10\) and the limits are from \(x = 0\) to \(x = 9\). This setup allows us to determine the total area under the curve from the left endpoint to the right. Definite integrals result in a numerical value, which can be positive, negative, or zero. The sign is important because if it's negative, it usually suggests that the curve is below the x-axis, as seen in this example. The value obtained reflects the signed area between the curve and the x-axis.

Finding the area under a curve is a common application of integration in calculus. This involves calculating the total space between a curve and the horizontal axis over a specified interval. To achieve this, we decompose the interval into infinitely small sub-intervals and sum the rectangles' areas formed by their adjacent sides:

• one side aligns with the curve,
• and the opposite side is on the x-axis,

using integrals. For the function \(y = \sqrt{x} - 10\), the region of interest is bounded by \(x = 0\) and \(x = 9\). Evaluating the integral gives the signed area of this region. Because the function has negative values within this interval (as it dips below the x-axis), the integral calculation produces a negative result. This negative result merely indicates the direction of the area in regard to the x-axis; understanding this fact can help avoid confusion when interpreting the result.

Sketching graphs is a vital skill in calculus, providing a visual understanding of mathematical functions. When asked to sketch a graph, we start by plotting key points and intercepts. For the function \(y = \sqrt{x} - 10\), the initial step is identifying where the curve intersects the x-axis. Setting \(\sqrt{x} - 10 = 0\), we solve for \(x\). However, as noted, for this exercise, the relevant range is from \(x = 0\) to \(x = 9\). The plotted graph will predominantly lie below the x-axis in this interval because \(\sqrt{x} - 10\) results in values less than zero. While sketching, it's helpful to:

• mark key points, like intersections with axes,
• identify symmetries or special shapes of the curve,
• and smoothly connect these points for accurate representation.

This visual approach assists in the verification of calculated areas by providing a holistic view of the curve's behavior over the designated interval.