Find The Area Between Y=2x^2 And Y=12x4x^2.
Understanding how to find the area between two curves is a fundamental concept in calculus, especially in the context of integration. This process involves determining the region enclosed by the two functions and calculating its area using definite integrals. In this article, we will explore the step-by-step approach to find the area between the curves y=2x^2 and y=12x4x^2, providing insights into the underlying principles, methods, and applications.
Introduction to Area Between Curves
The area between two curves is the region enclosed by the graphs of the functions over a specific interval. Typically, this involves:
- Identifying the points of intersection between the two curves.
- Determining which curve is on top (greater y-value) and which is on the bottom.
- Setting up an integral to calculate the area between the curves over the interval.
Calculating this area provides insights into the space occupied between the functions and is crucial in various fields such as physics, engineering, and economics.
Understanding the Given Functions
Before proceeding, let's examine the functions involved:
- First function: y = 2x^2
- Second function: y = 12x4x^2
At first glance, the second function y=12x4x^2 appears to be a typo or misrepresentation. Likely, the intended function is y=12x·4x^2, which simplifies to y=12x 4x^2. Alternatively, it could be y=12x^4x^2, but that seems less likely due to notation.
Assumption: The second function is y = 12x 4x^2.
Let's simplify y=12x 4x^2:
- y = 12x 4x^2 = (12 4) x x^2 = 48 x^3
Thus, the second function simplifies to y=48x^3.
Summary:
- y = 2x^2
- y = 48x^3
Note: If the original problem intended a different interpretation, please clarify. For now, we'll proceed with y=48x^3 as the second function.
Step 1: Find Intersection Points
The first step in calculating the area between the curves is to find their points of intersection. These are the x-values where y=2x^2 and y=48x^3 are equal:
2x^2 = 48x^3
Let's solve for x:
2x^2 = 48x^3
Divide both sides by 2:
x^2 = 24x^3
Divide both sides by x^2 (assuming x ≠ 0):
1 = 24x
Solve for x:
x = 1/24
Special case: x=0
Check if x=0 is an intersection:
At x=0:
- y=20=0
- y=480=0
Yes, x=0 is an intersection point.
Therefore, the solutions are:
- x=0
- x=1/24
Note: Since dividing by x^2 is valid only when x ≠ 0, we explicitly check x=0, which also satisfies the original equation.
---
Result: The curves intersect at x=0 and x=1/24.
---
Step 2: Determine Which Curve Is On Top
To set up the integral, we need to identify which function has the greater y-value in the interval between the intersection points.
Test a point between 0 and 1/24, say x=1/48:
- y=2x^2 = 2(1/48)^2 ≈ 2(1/2304) ≈ 0.000868
- y=48x^3 = 48(1/48)^3 = 48(1/110592) ≈ 0.000434
Since y=2x^2 > y=48x^3 at x=1/48, the parabola y=2x^2 is on top between 0 and 1/24.
As x approaches 0, both functions approach 0.
---
Conclusion: For x in [0, 1/24]:
- Top curve: y=2x^2
- Bottom curve: y=48x^3
Step 3: Set Up the Integral for the Area
The area \(A\) between the two curves over the interval [0, 1/24] is:
\[
A = \int_{x=0}^{x=1/24} [\text{top function} - \text{bottom function}] \, dx
\]
Substituting:
\[
A = \int_{0}^{1/24} [2x^2 - 48x^3] \, dx
\]
---
Step 4: Compute the Integral
Let's evaluate:
\[
A = \int{0}^{1/24} 2x^2 \, dx - \int{0}^{1/24} 48x^3 \, dx
\]
Calculate each separately:
- \(\int 2x^2 \, dx = 2 \times \frac{x^3}{3} = \frac{2}{3} x^3\)
- \(\int 48x^3 \, dx = 48 \times \frac{x^4}{4} = 12 x^4\)
Now, evaluate from 0 to 1/24:
\[
A = \left[ \frac{2}{3} x^3 \right]0^{1/24} - \left[ 12 x^4 \right]0^{1/24}
\]
Calculate each term:
- For \(x=1/24\):
\[
\frac{2}{3} \left(\frac{1}{24}\right)^3 = \frac{2}{3} \times \frac{1}{24^3} = \frac{2}{3} \times \frac{1}{13,824}
\]
- For \(x=1/24\):
\[
12 \left(\frac{1}{24}\right)^4 = 12 \times \frac{1}{24^4} = 12 \times \frac{1}{331,776}
\]
Calculate these fractions:
\[
A = \frac{2}{3} \times \frac{1}{13,824} - 12 \times \frac{1}{331,776}
\]
Simplify:
\[
A = \frac{2}{3 \times 13,824} - \frac{12}{331,776}
\]
Note that \(3 \times 13,824 = 41,472\).
So:
\[
A = \frac{2}{41,472} - \frac{12}{331,776}
\]
Express both with a common denominator:
- The denominators are 41,472 and 331,776.
Observe that:
\[
331,776 ÷ 41,472 = 8
\]
So, rewrite the first fraction with denominator 331,776:
\[
\frac{2}{41,472} = \frac{2 \times 8}{41,472 \times 8} = \frac{16}{331,776}
\]
Now, the area becomes:
\[
A = \frac{16}{331,776} - \frac{12}{331,776} = \frac{16 - 12}{331,776} = \frac{4}{331,776}
\]
Simplify numerator and denominator:
Divide numerator and denominator by 4:
\[
A = \frac{1}{83,044}
\]
(Actually, 331,776 ÷ 4 = 83,044).
Final Result:
\[
\boxed{
A = \frac{1}{83,044}
}
\]
This is the exact area between the curves over the interval from 0 to 1/24.
---
Summary and Final Remarks
- The functions involved were identified as y=2x^2 and y=48x^3, based on the provided expression.
- The intersection points are at x=0 and x=1/24.
- The region between the curves on [0, 1/24] was calculated via definite integrals.
- The resulting exact area is \(\frac{1}{83,044}\).
Applications of Finding Area Between Curves
Calculating the area between curves is a fundamental technique in various disciplines:
- Physics: Determining work done or energy stored in systems.
- Economics: Calculating consumer and producer surpluses.
- Engineering: Analyzing material stress regions.
- Biology: Estimating areas in biological models.
Understanding how to accurately compute these areas enables professionals to make informed decisions and perform precise analyses.
Additional Tips for Solving Similar Problems
- Always verify the functions and clarify any ambiguous notation.
- Find points of intersection first to determine the limits of integration.
- Sketch the graphs if possible, to visualize which curve is on top
