Problem 38 - Essential Calculus

Question

Sketch the region bounded by the graphs of the functions, and find the area of the region. $$ f(x)=\sec \frac{\pi x}{4} \tan \frac{\pi x}{4}, g(x)=(\sqrt{2}-4) x+4, \quad x=0 $$

Step-by-Step Solution

Answer

The area of the region can be calculated by setting up an integral \(\int_a^b (f(x) - g(x)) dx\) where \(a\) is the x-coordinate of the intersection point and \(b=0\). Calculate the integral.

Step 1: Sketch the functions

Start by plotting the two functions, \(f(x)=\sec \frac{\pi x}{4} \tan \frac{\pi x}{4}\) and \(g(x)=(\sqrt{2}-4) x + 4, \quad x=0\), on a graph. You can do this manually or use a graphing calculator or software.

Step 2: Find the intersection point

Find the x-value where \(f(x)\) intersects \(g(x)\). To do this, set \(f(x) = g(x)\) and solve for \(x\).

Step 3: Set up the integral

The area between two curves is given by \(\int (f(x) - g(x)) dx\) from the x value in \(x=0\) to the x-coordinate of the intersection point. So, set up an integral \(\int (f(x) - g(x)) dx\) from 0 to the x-coordinate of the intersection point.

Step 4: Calculate the integral

Calculate the integral using fundamental theorem of calculus or use appropriate numerical method if the integral is not solvable analytically

Key Concepts

Area between curves

When we talk about the area between curves, we're discussing the space sandwiched between two function graphs on a coordinate plane. To find this area, we first need to understand which function lies above the other over a specified interval. Once identified, we calculate the area between them by taking the difference of the two functions and integrating over that interval.
For this exercise, we have the functions \( f(x) = \sec \frac{\pi x}{4} \tan \frac{\pi x}{4} \) and \( g(x) = (\sqrt{2} - 4)x + 4 \). The area is determined by integrating the vertical distance between these curves from where they intersect back to the given boundary, such as \( x = 0 \).
This method requires setting up an integral that subtracts the lower function from the upper function across the domain where they intersect. The mathematical representation can be expressed as:

• \( A = \int_{a}^{b} (f(x) - g(x)) \, dx \)

This formula essentially calculates the net area by integrating the vertical distances across the bounds of interest.

Graph of functions

Graphs of functions are visual representations of equations where each input (or \(x\) value) has a corresponding output (or \(y\) value). Graphing allows us to see the relationship between the two variables and how they change together. It's a crucial tool for visualizing equations like \( f(x) \) and \( g(x) \) from the exercise.
To start, you might plot every point of \( f(x) = \sec \frac{\pi x}{4} \tan \frac{\pi x}{4} \) and \( g(x) = (\sqrt{2} - 4)x + 4 \), then connect these points to see the overall curve. Technological tools, such as graphing calculators or software, can make this task easier and more accurate.
By understanding and sketching these graphs, you can visually locate the area and intersection of the functions, which is essential for solving problems involving areas between curves and definite integrals.

Intersection of functions

Finding the intersection of functions is a key step when calculating areas between curves. An intersection occurs where two functions have equal \( y \)-values for the same \( x \)-value. In practical terms, this point tells us where one graph crosses another.
For this problem, finding the intersection involves setting \( f(x) = g(x) \) and solving this equation for \( x \). This requires finding the \( x \)-coordinate where both \( \sec \frac{\pi x}{4} \tan \frac{\pi x}{4} \) and \( (\sqrt{2} - 4)x + 4 \) yield the same result.
This intersection point also serves as the upper boundary for integrating when calculating the area. Therefore, it's crucial to identify this point accurately for setting up and solving the integral effectively.

Definite integrals

Definite integrals are used to calculate the accumulation of quantities, like areas under curves, between specified boundaries. In this case, for the area between the curves \( f(x) \) and \( g(x) \), we use the definite integral to find the exact size of the area between them from one point to another.
Having determined the functions' intersection, the definite integral is set from one boundary (\( x=0 \)) to another, which is the \( x \)-coordinate of intersection.
The integration process will subtract the value of \( g(x) \) from \( f(x) \) across this interval, portrayed as:

• \( \int_{0}^{\text{intersection \;point}} (f(x) - g(x)) \, dx \)

By solving this integral, we compute the accumulated area, offering a quantitative description of the space between the curves over the specified interval. Understanding how to set up and approximate or solve this integral is crucial for finding exact areas in calculus.