Problem 48 - Calculus Single Variable

Question

Calculate the area of the region between the pair of curves. $$ x=y \quad x=y^{4} $$

Step-by-Step Solution

Answer

The area between the curves is \( \frac{3}{10} \) square units.

Step 1: Understand the Problem

We need to find the area enclosed between the two curves: \( x = y \) and \( x = y^4 \). First, it's crucial to determine where these curves intersect as these intersection points will be the limits of integration when calculating the area.

Step 2: Find Points of Intersection

To find the points of intersection, set the equations equal to each other: \( y = y^4 \). Rearranging gives \( y^4 - y = 0 \), which factors to \( y(y^3 - 1) = 0 \). Solving this gives \( y = 0 \) or \( y^3 = 1 \), thus \( y = 1 \). Intersection points are at \( y = 0 \) and \( y = 1 \).

Step 3: Set Up the Integral

The area between the curves from \( y = 0 \) to \( y = 1 \) can be found using the integral: \[ A = \int_{0}^{1} (y - y^4) \, dy \]Here, \( y \) represents the curve \( x = y \) and \( y^4 \) represents the curve \( x = y^4 \).

Step 4: Evaluate the Integral

Evaluate the integral \[ A = \int_{0}^{1} (y - y^4) \, dy \]This simplifies to: \[ A = \left[ \frac{y^2}{2} - \frac{y^5}{5} \right]_{0}^{1} \]Calculate the values: \[ = \left( \frac{1^2}{2} - \frac{1^5}{5} \right) - \left( \frac{0^2}{2} - \frac{0^5}{5} \right) \]\[ = \frac{1}{2} - \frac{1}{5} \]Simplify: \[ = \frac{5}{10} - \frac{2}{10} = \frac{3}{10} \].

Step 5: Conclusion

The calculated area of the region between the two curves \( x = y \) and \( x = y^4 \) from \( y = 0 \) to \( y = 1 \) is \( \frac{3}{10} \) square units.

Key Concepts

Intersection Points

To find the area between two curves, it's essential to first find their intersection points. This is where the curves meet each other, serving as critical markers in setting the boundaries for our calculations. For the curves given, we have the equations \( x = y \) and \( x = y^4 \). To find their intersection points, we set these equations equal to each other: \( y = y^4 \).

Solving this equation involves rearranging it into the form \( y^4 - y = 0 \). Factoring out a \( y \), we get \( y(y^3 - 1) = 0 \). From here, we can easily see the solutions: either \( y = 0 \) or \( y^3 = 1 \). Solving \( y^3 = 1 \) gives \( y = 1 \). Therefore, the intersection points are at \( y = 0 \) and \( y = 1 \). These points will act as the limits of integration in our next steps.

Integral Calculation

Once we've identified the intersection points, the next step is to set up an integral. This integral will help us calculate the area between the curves. Starting with our intersection points \( y = 0 \) and \( y = 1 \), we know these define the region over which we need to integrate.

The integral that represents the area between these curves is given by:

• \( A = \int_{0}^{1} (y - y^4) \, dy \)

Here, \( y \) corresponds to the line \( x = y \), which is the upper function, and \( y^4 \) corresponds to \( x = y^4 \), the lower function. The integral thus evaluates the difference between these two functions over the interval from 0 to 1.

Limits of Integration

When calculating definite integrals, the limits of integration are crucial. They define the exact interval over which the area is being calculated. In this exercise, the limits come directly from the intersection points found earlier: \( y = 0 \) and \( y = 1 \).

These limits tell us that the region of interest extends from 0 to 1 along the \( y \)-axis. It's important to accurately identify these limits because they directly affect the area you compute. By evaluating the integral within these bounds, rather than over an entire function, you target only the region enclosed between the curves, leading to:

• \( A = \left[ \frac{y^2}{2} - \frac{y^5}{5} \right]_{0}^{1} \)

Substituting these limits into this evaluated expression gives us the area we sought to determine between the curves.