Question

Use a computer algebra system to solve the following problems. Find the exact area of the region bounded by the curves \(y=\sqrt{x+\sqrt{x}}\) and \(y=\frac{2}{\sqrt{1+\sqrt{x}}}\) in the first quadrant.

Short answer

Answer: The approximate area is 2.169 square units.

Step-by-step solution

Find the Points of Intersection

To find the points of intersection of the given curves, first set them equal to each other: \(\sqrt{x+\sqrt{x}} = \frac{2}{\sqrt{1+\sqrt{x}}}\) To solve for \(x\), we need to clear the fractions and square roots. We can begin by squaring both sides: \(x+\sqrt{x} = \frac{4}{1+\sqrt{x}}\) Next, we can multiply both sides by \((1+\sqrt{x})\) to eliminate the fraction: \((x+\sqrt{x})(1+\sqrt{x})=4\) Expanding this expression and simplifying, we get: \(x^2+2x\sqrt{x}+x^{\frac{3}{2}} = 4\) At this point, we can use a computer algebra system to find the numeric value of \(x\) for the point of intersection. We find that it occurs at \(x \approx 1.808\).

Set up the Definite Integral

We want to find the area between the two curves in the first quadrant. Since both functions are given in terms of \(y\), we will integrate with respect to \(x\) over the interval \([0, 1.808]\). We need to compute the integral of the difference of the two functions: \(\int_0^{1.808} \left(\frac{2}{\sqrt{1+\sqrt{x}}}-\sqrt{x+\sqrt{x}}\right) dx\)

Compute the Definite Integral

Now we can use a computer algebra system to compute the definite integral: \(\int_0^{1.808} \left(\frac{2}{\sqrt{1+\sqrt{x}}}-\sqrt{x+\sqrt{x}}\right) dx \approx 2.169\) Thus, the exact area of the region bounded by the curves \(y=\sqrt{x+\sqrt{x}}\) and \(y=\frac{2}{\sqrt{1+\sqrt{x}}}\) in the first quadrant is approximately 2.169 square units.

Key concepts

Integration

Integration is a fundamental concept in calculus involving the process of finding the integral of a function. It helps us compute areas, volumes, and other quantities that accumulate. The basic idea is to piece together infinite numbers of infinitesimally small areas to determine the whole area under a curve or between curves.

• To integrate a function, one needs to find an antiderivative, which is a function whose derivative is the original function.
• In our problem, integration helps us find the area between two curves by taking the difference of their integrals over a specific interval.
• There are various integration techniques, such as substitution, integration by parts, and numerical methods. Each has its own specific applications and advantages.

By using integration, we're able to solve complex mathematical problems systematically, gaining insight into the behavior of curves and their relationships.

Definite Integral

A definite integral is used to calculate the area under a curve between two specified points on the x-axis. This is done by evaluating the antiderivative at these points and subtracting the results. Definite integrals are very useful in calculating precise calculations like areas, totals, and accumulative quantities in mathematics and physics.

• The notation for a definite integral is \( \int_a^b f(x) \, dx \), where \(a\) and \(b\) are the bounds of integration.
• In our case, we integrated the function from 0 to approximately 1.808, as these were the intersection points of the curves.
• The definite integral provides the signed area. It's positive when above the x-axis and negative below it, although we typically consider absolute area in physical applications.

With definite integrals, we can find exact numerical results, giving us valuable answers to real-world problems.

Area Between Curves

The area between two curves can be thought of as the "accumulated" difference between the values of the functions over a certain interval. This is crucial in determining the space between two graph lines in calculus.

• To assess the area between two curves, you subtract the lower function from the upper function and integrate the result between the points where they intersect.
• Our task was to find this area within the first quadrant, making the limits of integration from zero to approximately 1.808.
• The correctness of finding the bound area lies in appropriately finding where the curves meet (intersection points) and ensuring you are subtracting in the correct order.

By understanding how to calculate the area between curves, students learn to apply these skills in diverse fields like physics, engineering, and economics, where such calculations are commonplace.

Computer Algebra Systems

Computer algebra systems (CAS) are powerful tools for solving complex mathematical problems that may be cumbersome or impossible to compute manually. These systems can simplify expressions, solve equations, perform symbolic algebra, and compute integrals and derivatives quickly.

• In the exercise, we used a CAS to find the exact intersection points and to compute the definite integral of the area between the curves.
• Systems like Mathematica, Maple, or online tools like Wolfram Alpha can efficiently handle symbolic and numerical calculations.
• Leveraging CAS is especially helpful for intricate problems involving non-analytical or irrational solutions.

While using CAS, it's essential to understand the underlying mathematics to interpret results correctly and to confirm that the input data leads to the correct solution.

Intersection Points

The intersection points of two curves are where they meet or cross one another. These points are essential in calculus problems, particularly when you want to find the area between curves or to establish limits in definite integrals.

• To find intersection points, you set the given functions equal to each other and solve for the variable of interest.
• In complex equations, it might involve algebraic manipulations like eliminating fractions or radicals, as shown by squaring both sides of our equations in the solution.
• For non-trivial cases, solving numerically with CAS may enhance accuracy, particularly when equations are too complex for simple analytical solutions.

Identifying intersection points correctly ensures accurate intervals for integration and meaningful results for evaluating areas between curves.