Set up and evaluate the definite integral that gives the...

Set up and evaluate the definite integral that gives the area of the region bounded by the graph of the function and the tangent line to the graph at the given point.
$$
y=\frac{2}{1+4 x^{2}}, \quad\left(\frac{1}{2}, 1\right)
$$

Key Concepts

Finding the Equation of the Tangent Line

To determine the equation of a tangent line, one must first compute the derivative of the function to obtain the slope at the point of tangency. Then, using the point-slope form of a line, the tangent line's equation can be formed, which exactly matches the function's rate of change at that point.

Setting Up the Definite Integral for Area

Once the equation of the tangent line and the points where it intersects the curve are determined, these intersection points serve as the limits of integration. The definite integral is then set up by integrating the difference between the function and the tangent line (taking care to subtract the lower function from the upper function) over this interval to compute the bounded area.

Tangent Line

A tangent line to a curve at a particular point is a straight line that touches the curve without cutting across it in that immediate vicinity. It represents the instantaneous rate of change of the function at that point, serving as a linear approximation to the curve locally.

Definite Integral

A definite integral computes the accumulated sum of values of a function over a specific interval. It is often used to calculate the area under a curve between two bounds, providing a precise measure of the net 'accumulated' quantity, which in many cases is used to determine areas, volumes, and other physical quantities.

Area Between Curves

The area between curves is found by integrating the difference between two functions over an interval defined by their intersection points. This involves subtracting the lower function from the upper function at every point in the interval, which yields the exact area of the region bounded by the curves.

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