# Area of Region R Bounded by Line and Curve

• Samuelb88
In summary: This equation represents a line with slope (f'(x_i)) and y-intercept (f(x_i)). Now, if we consider the slope of this tangent line at any point (x_i, f(x_i)), we can write it as:m_i = f'(x_i) = \frac{df}{dx}|_(x_i)This means that the slope of the tangent line at any point is equal to the slope of the curve at that point. Now, using this relationship, we can write the equation of the line perpendicular to the line y = mx + b at the point (x_i, f(x_i)) as:y = -\frac
Samuelb88

## Homework Statement

Let C be the arc of the curve y = f(x) between points P(p,f(p)), Q(q,f(q)), and let R be the region bounded by C, the line y = mx + b, and the perpendiculars by the line from P and Q.

http://img15.imageshack.us/img15/6827/pic1ds.jpg

Show that the area R is

$$\frac{1}{1+m^2}\right) \int_q^p (f(x) - mx - b)(1+m\frac{df}{dx}\right) ) dx$$

Hint: This formula can be verified by subtracting areas, but it will be helpful to derive region R using rectangles perpendicular to the line, shown in the figure below.

http://img163.imageshack.us/img163/1365/pic2c.jpg

## The Attempt at a Solution

Tangent to C @ (x_i,f(x_i)): $$y=f'(x_i)(x-x_i) + f(x_i)$$

I let L equal the line segment "?" perpendicular to the line y=mx+b and used the distance equation to determine its value.

$$L = ((x_i-x)^2+(f'(x_i)(x-x_i) + f(x_i) - mx - b)^2)^(^1^/^2^)$$

And the area A

A = L(change in u)

So the region R should equal the limit of all the approximating rectangles.

Pretty confused as of how to proceed. Any suggestions would be greatly appreciated. :)

Last edited by a moderator:

Thank you for your post. It seems like you have made some good progress in understanding the problem and determining the necessary equations. However, I would suggest taking a step back and looking at the problem from a more general perspective.

Firstly, it is important to understand what the question is asking for. The problem states that we want to find the area of the region R, which is bounded by the curve y = f(x), the line y = mx + b, and the perpendiculars from points P and Q. This means that we are looking for the integral of the function (f(x) - mx - b) over the interval [q,p]. This is because the area of a region can be calculated by finding the integral of the function that represents the boundary of the region.

Now, the hint given in the problem suggests using rectangles perpendicular to the line y = mx + b to approximate the region R. This means that we can divide the region R into smaller rectangles, each with a height of (f(x) - mx - b) and a width of (dx). The sum of the areas of these rectangles will be an approximation of the total area of R. To get a more accurate approximation, we can increase the number of rectangles, making their width (dx) smaller and smaller.

Using this approach, we can write the area of R as the limit of the sum of the areas of these rectangles as the width (dx) approaches 0. This is the same as taking the integral of the function (f(x) - mx - b) over the interval [q,p].

Therefore, the area of R can be expressed as:

A = ∫q^p (f(x) - mx - b) dx

However, this is not the final answer as the problem asks us to show that the area of R is equal to:

\frac{1}{1+m^2}\right) \int_q^p (f(x) - mx - b)(1+m\frac{df}{dx}\right) ) dx

To prove this, we need to use the relationship between the slope of the curve (df/dx) and the slope of the line (m) to show that the above expression is equivalent to the one we derived earlier. I would suggest starting by writing the equation of the tangent line to the curve y = f(x) at any point (x_i, f(x_i)) as:

y =

## 1. What does the "area of region R bounded by line and curve" refer to?

The "area of region R bounded by line and curve" refers to the measure of the two-dimensional space enclosed by a straight line and a curved line or shape.

## 2. How is the area of region R calculated?

The area of region R can be calculated by using mathematical formulas, such as integration, that involve the coordinates of the line and curve. It can also be approximated using numerical methods or by breaking the region into smaller, simpler shapes and adding their areas together.

## 3. What is the significance of calculating the area of region R?

Calculating the area of region R is important in many fields, including mathematics, engineering, and physics. It can be used to solve real-world problems, such as finding the area of a plot of land or the volume of a 3D object. It also helps in understanding and analyzing the properties of different shapes and curves.

## 4. Can the area of region R be negative?

No, the area of region R cannot be negative. It represents a physical quantity, and a negative area does not have any real-world meaning. If the result of a calculation is negative, it indicates that the line and curve intersect and the region is actually divided into two parts with opposite areas.

## 5. Are there any limitations to calculating the area of region R?

Yes, there are some limitations to calculating the area of region R. For instance, the line and curve must be well-defined and continuous, and the region should be bounded and have a finite area. In some cases, the calculation may also involve complex mathematical concepts that require advanced knowledge and techniques.

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