Finding Area Under Curve Bounded By Both X And Y Axis.

=====================================================

Introduction

---

In real analysis, integration is a fundamental concept used to find the area under curves. Given a function $f(x)$, the area under the curve can be calculated using definite integrals. In this article, we will discuss how to find the area under the curve $f(x) = C - x^4$ bounded by the $x$ and $y$ axes, where $x$ is any real positive number.

Understanding the Problem

---

To find the area under the curve $f(x) = C - x^4$, we need to evaluate the definite integral of the function from $x = 0$ to $x = a$, where $a$ is a positive real number. The area under the curve can be represented as:

$$A = \int_{0}^{a} (C - x^4) dx$$

Evaluating the Definite Integral

---

To evaluate the definite integral, we can use the power rule of integration, which states that:

$$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$

Using this rule, we can integrate the function $f(x) = C - x^4$ as follows:

$$\int (C - x^4) dx = Cx - \frac{x^5}{5} + C$$

Applying the Fundamental Theorem of Calculus

---

The fundamental theorem of calculus states that the definite integral of a function $f(x)$ from $x = a$ to $x = b$ can be evaluated as:

$$\int_{a}^{b} f(x) dx = F(b) - F(a)$$

where $F(x)$ is the antiderivative of $f(x)$. In this case, the antiderivative of $f(x) = C - x^4$ is:

$$F(x) = Cx - \frac{x^5}{5} + C$$

Finding the Area under the Curve

---

Now that we have the antiderivative of the function, we can evaluate the definite integral from $x = 0$ to $x = a$ as follows:

$$A = \int_{0}^{a} (C - x^4) dx = F(a) - F(0)$$

Substituting the values of $F(a)$ and $F(0)$, we get:

$$A = \left(Ca - \frac{a^5}{5} + C\right) - \left(C(0) - \frac{0^5}{5} + C\right)$$

Simplifying the expression, we get:

$$A = Ca - \frac{a^5}{5}$$

Conclusion

---

In this article, we discussed how to find the area under the curve $f(x) = C - x^4$ bounded by the $x$ and $y$ axes, where $x$ is any real positive number. We used the power rule of integration and the fundamental theorem of calculus to evaluate the definite integral and find the area under the curve. The final expression for the area under the curve is:

$$A = Ca - \frac{a^5}{5}$$

This result can be used to find the area under similar curves bounded by the $x$ and $y$ axes.

Future Work--------------

In future work, we can explore other applications of integration, such as finding the area under curves bounded by multiple axes or finding the volume of solids of revolution. We can also investigate the use of numerical methods to approximate the area under curves when the antiderivative is not available.

References

---

• [1] "Calculus" by Michael Spivak
• [2] "Introduction to Real Analysis" by Bartle and Sherbert
• [3] "Calculus: Early Transcendentals" by James Stewart

Code

---

import sympy as sp
x = sp.symbols('x')
f = C - x**4

A = sp.integrate(f, (x, 0, a))

print(A)

Note: The code is written in Python using the SymPy library, which is a powerful tool for symbolic mathematics. The code defines the variable and function, evaluates the definite integral, and prints the result.

=====================================================

Introduction

---

In our previous article, we discussed how to find the area under the curve $f(x) = C - x^4$ bounded by the $x$ and $y$ axes, where $x$ is any real positive number. In this article, we will answer some frequently asked questions related to finding the area under curves bounded by the $x$ and $y$ axes.

Q&A

---

Q: What is the area under the curve $f(x) = C - x^4$ bounded by the $x$ and $y$ axes?

A: The area under the curve $f(x) = C - x^4$ bounded by the $x$ and $y$ axes is given by the definite integral:

$$A = \int_{0}^{a} (C - x^4) dx = Ca - \frac{a^5}{5}$$

Q: How do I find the area under a curve bounded by the $x$ and $y$ axes?

A: To find the area under a curve bounded by the $x$ and $y$ axes, you need to evaluate the definite integral of the function from $x = 0$ to $x = a$, where $a$ is a positive real number.

Q: What is the fundamental theorem of calculus?

A: The fundamental theorem of calculus states that the definite integral of a function $f(x)$ from $x = a$ to $x = b$ can be evaluated as:

$$\int_{a}^{b} f(x) dx = F(b) - F(a)$$

where $F(x)$ is the antiderivative of $f(x)$.

Q: How do I find the antiderivative of a function?

A: To find the antiderivative of a function, you need to integrate the function with respect to the variable. For example, the antiderivative of $f(x) = C - x^4$ is:

$$F(x) = Cx - \frac{x^5}{5} + C$$

Q: What is the power rule of integration?

A: The power rule of integration states that:

$$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$

Q: Can I use numerical methods to approximate the area under a curve?

A: Yes, you can use numerical methods to approximate the area under a curve when the antiderivative is not available. Some common numerical methods include the Riemann sum and the trapezoidal rule.

Conclusion

---

In this article, we answered some frequently asked questions related to finding the area under curves bounded by the $x$ and $y$ axes. We discussed the fundamental theorem of calculus, the power rule of integration, and numerical methods for approximating the area under a curve.

Future Work--------------

In future work, we can explore other applications of integration, such as finding the area under curves bounded by multiple axes or finding the volume of solids of revolution. We can also investigate the use of numerical methods to approximate the area under curves when the antiderivative is not available.

References

---

• [1] "Calculus" by Michael Spivak
• [2] "Introduction to Real Analysis" by Bartle and Sherbert [3] "Calculus: Early Transcendentals" by James Stewart

Code

---

import sympy as sp

x = sp.symbols('x')
f = C - x**4

A = sp.integrate(f, (x, 0, a))

print(A)

Note: The code is written in Python using the SymPy library, which is a powerful tool for symbolic mathematics. The code defines the variable and function, evaluates the definite integral, and prints the result.

Common Mistakes

---

• Forgetting to evaluate the definite integral from $x = 0$ to $x = a$.
• Not using the fundamental theorem of calculus to evaluate the definite integral.
• Not using the power rule of integration to find the antiderivative of a function.
• Not using numerical methods to approximate the area under a curve when the antiderivative is not available.

Tips and Tricks

---

• Use the fundamental theorem of calculus to evaluate the definite integral.
• Use the power rule of integration to find the antiderivative of a function.
• Use numerical methods to approximate the area under a curve when the antiderivative is not available.
• Check your work by plugging in values for $a$ and $C$ to see if the result makes sense.