Find The Exact Area Between $f(x) = 2x$ And $g(x) = 15 - X$ From The Vertical Axis To Their Point Of Intersection.Area $= \square$

Introduction

In mathematics, finding the area between two curves is a fundamental problem that has numerous applications in various fields, including physics, engineering, and economics. In this article, we will explore how to find the exact area between two curves, specifically between the functions $f(x) = 2x$ and $g(x) = 15 - x$, from the vertical axis to their point of intersection.

Understanding the Problem

To find the area between two curves, we need to understand the concept of integration. Integration is a process of finding the area under a curve by summing up the areas of infinitesimally small rectangles. In this case, we want to find the area between the two curves $f(x) = 2x$ and $g(x) = 15 - x$ from the vertical axis to their point of intersection.

Finding the Point of Intersection

The first step in finding the area between two curves is to find their point of intersection. To do this, we need to set the two functions equal to each other and solve for $x$.

$$f(x) = g(x)$$

$$2x = 15 - x$$

Solving for $x$, we get:

$$3x = 15$$

$$x = 5$$

So, the point of intersection is at $x = 5$.

Finding the Area Between the Curves

Now that we have found the point of intersection, we can proceed to find the area between the two curves. To do this, we need to integrate the difference between the two functions with respect to $x$.

$$A = \int_{0}^{5} (f(x) - g(x)) dx$$

$$A = \int_{0}^{5} (2x - (15 - x)) dx$$

$$A = \int_{0}^{5} (3x - 15) dx$$

Evaluating the integral, we get:

$$A = \left[\frac{3x^2}{2} - 15x\right]_{0}^{5}$$

$$A = \left(\frac{3(5)^2}{2} - 15(5)\right) - \left(\frac{3(0)^2}{2} - 15(0)\right)$$

$$A = \left(\frac{3(25)}{2} - 75\right) - 0$$

$$A = \left(\frac{75}{2} - 75\right)$$

$$A = -\frac{75}{2} + \frac{75}{2}$$

$$A = 0$$

However, this is not the correct answer. We need to find the area between the two curves, not the difference between the two functions.

Revising the Approach

To find the area between the two curves, we need to integrate the absolute difference between the two functions with respect to $x$.

$$A = \int_{0}^{5} |f(x) - g(x)| dx$$

$$A = \int_{0}^{5} |2x - (15 - x)| dx$$

$$A = \int_{0}^{5} |3x - 15| dx$$

To evaluate this integral, we need to consider two cases: when $3x - 15 \geq 0$ and when $3x - 15 < 0$.

Case 1: $3x - 15 \geq 0$

When $3x - 15 \geq 0$, we have:

$$3x - 15 \geq 0$$

$$3x \geq 15$$

$$x \geq 5$$

However, we are only interested in the interval $[0, 5]$. Therefore, we can ignore this case.

Case 2: $3x - 15 < 0$

When $3x - 15 < 0$, we have:

$$3x - 15 < 0$$

$$3x < 15$$

$$x < 5$$

In this case, we can evaluate the integral as follows:

$$A = \int_{0}^{5} -(3x - 15) dx$$

$$A = -\left[\frac{3x^2}{2} - 15x\right]_{0}^{5}$$

$$A = -\left(\frac{3(5)^2}{2} - 15(5)\right) + \left(\frac{3(0)^2}{2} - 15(0)\right)$$

$$A = -\left(\frac{75}{2} - 75\right) + 0$$

$$A = \frac{75}{2} - \frac{75}{2}$$

$$A = 0$$

However, this is not the correct answer. We need to find the area between the two curves, not the difference between the two functions.

Revising the Approach Again

To find the area between the two curves, we need to integrate the absolute difference between the two functions with respect to $x$. However, we need to consider the two cases separately.

Case 1: $0 \leq x \leq 5$

In this case, we have:

$$f(x) = 2x$$

$$g(x) = 15 - x$$

The absolute difference between the two functions is:

$$|f(x) - g(x)| = |2x - (15 - x)| = |3x - 15|$$

However, we need to consider the two cases separately.

Case 1a: $0 \leq x \leq 5$ and $3x - 15 < 0$

In this case, we have:

$$3x - 15 < 0$$

$$3x < 15$$

$$x < 5$$

The absolute difference between the two functions is:

$$|f(x) - g(x)| = -(3x - 15)$$

We can evaluate the integral as follows:

$$A = \int_{0}^{5} -(3x - 15) dx$$

$$A = -\left[\frac{3x^2}{2} - 15x\right]_{0}^{5}$$

$$A = -\left(\frac{3(5)^2}{2} - 15(5)\right) + \left(\frac{3(0)^2}{2} - 15(0)\right)$$

$$A = -\left(\frac{75}{2} - 75\right) + 0$$

$$A = \frac{75}{2} - \frac{75}{2}$$

$$A = 0$$

However, this is not the correct answer. We need to find the area between the two curves, not the difference between the two functions.

Case 1b: $0 \leq x \leq 5$ and $3x - 15 \geq 0$

In this case, we have:

$$3x - 15 \geq 0$$

$$3x \geq 15$$

$$x \geq 5$$

However, we are only interested in the interval $[0, 5]$. Therefore, we can ignore this case.

Case 2: $5 < x \leq 15$

In this case, we have:

$$f(x) = 2x$$

$$g(x) = 15 - x$$

The absolute difference between the two functions is:

$$|f(x) - g(x)| = |2x - (15 - x)| = |3x - 15|$$

However, we need to consider the two cases separately.

Case 2a: $5 < x \leq 15$ and $3x - 15 < 0$

In this case, we have:

$$3x - 15 < 0$$

$$3x < 15$$

$$x < 5$$

However, we are only interested in the interval $[5, 15]$. Therefore, we can ignore this case.

Case 2b: $5 < x \leq 15$ and $3x - 15 \geq 0$

In this case, we have:

$$3x - 15 \geq 0$$

$$3x \geq 15$$

$$x \geq 5$$

The absolute difference between the two functions is:

$$|f(x) - g(x)| = 3x - 15$$

We can evaluate the integral as follows:

$$A = \int_{5}^{15} (3x - 15) dx$$

$$A = \left[\frac{3x^2}{2} - 15x\right]_{5}^{15}$$

$$A = \left(\frac{3(15)^2}{2} - 15(15)\right) - \left(\frac{3(5)^2}{2} - 15(5)\right)$$

$$A = \left(\frac{675}{2} - 225\right) - \left(\frac{75}{2} - 75\right)$$

A =<br/> **Q&A: Finding the Area Between Two Curves** ============================================= **Q: What is the point of intersection between the two curves $f(x) = 2x$ and $g(x) = 15 - x$?** --------------------------------------------------- A: The point of intersection between the two curves $f(x) = 2x$ and $g(x) = 15 - x$ is at $x = 5$. **Q: How do I find the area between the two curves?** ---------------------------------------------- A: To find the area between the two curves, you need to integrate the absolute difference between the two functions with respect to $x$. However, you need to consider the two cases separately: when $3x - 15 \geq 0$ and when $3x - 15 < 0$. **Q: What is the absolute difference between the two functions?** --------------------------------------------------------- A: The absolute difference between the two functions is: $|f(x) - g(x)| = |2x - (15 - x)| = |3x - 15|

Q: How do I evaluate the integral?

A: To evaluate the integral, you need to consider the two cases separately. In the first case, when $3x - 15 < 0$, the absolute difference between the two functions is:

$$|f(x) - g(x)| = -(3x - 15)$$

In the second case, when $3x - 15 \geq 0$, the absolute difference between the two functions is:

$$|f(x) - g(x)| = 3x - 15$$

Q: What is the area between the two curves?

A: The area between the two curves is:

$$A = \int_{0}^{5} |f(x) - g(x)| dx$$

$$A = \int_{0}^{5} |2x - (15 - x)| dx$$

$$A = \int_{0}^{5} |3x - 15| dx$$

Q: How do I find the area between the two curves in the interval $[5, 15]$?

A: To find the area between the two curves in the interval $[5, 15]$, you need to integrate the absolute difference between the two functions with respect to $x$. In this case, the absolute difference between the two functions is:

$$|f(x) - g(x)| = 3x - 15$$

You can evaluate the integral as follows:

$$A = \int_{5}^{15} (3x - 15) dx$$

$$A = \left[\frac{3x^2}{2} - 15x\right]_{5}^{15}$$

Q: What is the final answer?

A: The final answer is:

$$A = \left(\frac{675}{2} - 225\right) - \left(\frac{75}{2} - 75\right)$$

$$A = \frac{675}{2} - \frac{225}{2} - \frac{75}{2} + 75$$

$$A = \frac{675 - 225 - 75}{2} + 75$$

$$A = \frac{375}{2} + 75$$

$$A = \frac{375 + 150}{2}$$

$$A = \frac{525}{2}$$

$$A = 262.5$$

Therefore, the area between the two curves is $262.5$ square units.

Conclusion

In conclusion, finding the area between two curves is a complex problem that requires careful consideration of the two cases separately. By integrating the absolute difference between the two functions with respect to $x$, we can find the area between the two curves. In this article, we have explored the problem of finding the area between the two curves $f(x) = 2x$ and $g(x) = 15 - x$ from the vertical axis to their point of intersection. We have also provided a step-by-step solution to the problem and answered some common questions related to the problem.