Introduction

In this article, we will walk through the process of finding the area of the region bounded by the curves ( y frac{x^2}{4} ) and ( y frac{x}{22} ). This problem involves basic calculus concepts like integration and the intersection of functions. We will employ step-by-step methods to solve this homework question and provide a clear explanation of the calculations involved.

Solving the Problem

Let's begin by understanding the problem at hand. We need to find the area of the region bounded by the functions ( y frac{x^2}{4} ) and ( y frac{x}{22} ). This task can be achieved by using the principles of calculus, specifically integration.

Step 1: Finding the Intersection Points

The first step is to find the points of intersection between the two curves. This is done by setting the two functions equal to each other:

[ frac{x^2}{4} frac{x}{22} ]

Multiplying both sides by 88 to clear the denominators:

[ 22x^2 4x ]

Rearrange the equation:

[ 22x^2 - 4x 0 ]

Factor out (x):

[ x(22x - 4) 0 ]

Solve for (x):

[ x 0 quad text{or} quad x frac{2}{11} ]

So, the points of intersection are ( x 0 ) and ( x frac{2}{11} ).

Step 2: Subtracting the Functions

Subtract the function ( y frac{x}{22} ) from ( y frac{x^2}{4} ). This "moves" the space between the two original curves down to the x-axis but does not change the area. The resulting function is:

[ frac{x^2}{4} - frac{x}{22} ]

Combine terms over a common denominator:

[ frac{22x^2 - 4x}{88} ]

Factor out (x):

[ frac{x(22x - 4)}{88} ]

Step 3: Calculating the Primitive Function

The next step is to calculate the primitive (antiderivative) of the function obtained in step 2:

[ int frac{x(22x - 4)}{88} dx ]

First, simplify the integrand:

[ frac{1}{88} int (22x^2 - 4x) dx ]

Integrate term by term:

[ frac{1}{88} left( frac{22x^3}{3} - 2x^2 right) C ]

So, the primitive function is:

[ frac{11x^3}{132} - frac{x^2}{44} C ]

Note that the constant (C) is not needed for the definite integral.

Step 4: Calculating the Definite Integral

To find the area, we need to evaluate the primitive function at the points of intersection and take the difference:

[ left[ frac{11x^3}{132} - frac{x^2}{44} right]_0^{frac{2}{11}} ]

Evaluate the function at ( x 0 ):

[ frac{11(0)^3}{132} - frac{(0)^2}{44} 0 ]

Evaluate the function at ( x frac{2}{11} ):

[ frac{11 left( frac{2}{11} right)^3}{132} - frac{left( frac{2}{11} right)^2}{44} ]

Simplify the expression:

[ frac{11 cdot frac{8}{1331}}{132} - frac{frac{4}{121}}{44} ] [ frac{88}{1331 cdot 132} - frac{4}{121 cdot 44} ] [ frac{88}{174248} - frac{1}{121} ] [ frac{88}{174248} - frac{1441}{174248} ] [ frac{88 - 1441}{174248} frac{-1353}{174248} ]

Since the area cannot be negative, take the absolute value:

[ frac{1353}{174248} approx 0.00775 ]

Hence, the area of the region bounded by ( y frac{x^2}{4} ) and ( y frac{x}{22} ) is approximately ( 0.00775 ) square units.

Conclusion

The process of finding the area of the region bounded by the curves ( y frac{x^2}{4} ) and ( y frac{x}{22} ) involves identifying the points of intersection, subtracting the functions, calculating the primitive function, and evaluating the definite integral. This method is a fundamental concept in calculus and is applicable to similar problems involving bounded regions.