Apply the Fundamental Theorem of Calculus

Oct 13, 2025 by miodeya.com 110 views

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Introduction

In this article, we will evaluate the given definite integral $\int_0^4\left(9 e^{0.25 x}+6 x\right) d x$ and express the answer in its simplest form. The definite integral is a fundamental concept in calculus, and it has numerous applications in various fields such as physics, engineering, and economics.

What is a Definite Integral?

A definite integral is a mathematical operation that calculates the area under a curve or the accumulation of a quantity over a given interval. It is denoted by the symbol $\int_a^b f(x) dx$ , where $f(x)$ is the function being integrated, and $a$ and $b$ are the limits of integration.

Evaluating the Definite Integral

To evaluate the given definite integral, we will use the following steps:

Integrate the first term: We will integrate the first term $9 e^{0.25 x}$ using the formula for the integral of an exponential function.
Integrate the second term: We will integrate the second term $6 x$ using the formula for the integral of a linear function.
Apply the Fundamental Theorem of Calculus: We will apply the Fundamental Theorem of Calculus to evaluate the definite integral.

Integrate the First Term

The first term is $9 e^{0.25 x}$ . To integrate this term, we will use the formula for the integral of an exponential function:

\int e^{ax} dx = \frac{1}{a} e^{ax} + C

where $a$ is a constant. In this case, $a = 0.25$ . Therefore, the integral of the first term is:

\int 9 e^{0.25 x} dx = 9 \left( \frac{1}{0.25} e^{0.25 x} \right) + C

Simplifying the expression, we get:

\int 9 e^{0.25 x} dx = 36 e^{0.25 x} + C

Integrate the Second Term

The second term is $6 x$ . To integrate this term, we will use the formula for the integral of a linear function:

\int x dx = \frac{x^2}{2} + C

Therefore, the integral of the second term is:

\int 6 x dx = 6 \left( \frac{x^2}{2} \right) + C

Simplifying the expression, we get:

\int 6 x dx = 3 x^2 + C

Apply the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus states that the definite integral of a function $f(x)$ from $a$ to $b$ is equal to the antiderivative of $f(x)$ evaluated at $b$ minus the antiderivative of $f(x)$ evaluated at $a$ . In this case, the antiderivative of the first term is $36 e^{0.25 x}$ , and the antiderivative of the second term is $3 x^2$ .

Therefore, the definite integral is:

\int_0^4\left(9 e^{0.25 x}+6 x\right) d x = \left[ 36 e^{0.25 x} + 3 x^2 \right]_0^4

Evaluating the expression at the limits of integration, we get:

\int_0^4\left(9 e^{0.25 x}+6 x\right) d x = \left( 36 e^{0.25 (4)} + 3 (4)^2 \right) - \left( 36 e^{0.25 (0)} + 3 (0)^2 \right)

Simplifying the expression, we get:

\int_0^4\left(9 e^{0.25 x}+6 x\right) d x = \left( 36 e^{1} + 48 \right) - \left( 36 e^{0} + 0 \right)

Since $e^0 = 1$ , we can simplify the expression further:

\int_0^4\left(9 e^{0.25 x}+6 x\right) d x = \left( 36 e^{1} + 48 \right) - 36

Simplifying the expression, we get:

\int_0^4\left(9 e^{0.25 x}+6 x\right) d x = 36 e^{1} + 12

Since $e^1 \approx 2.71828$ , we can approximate the value of the definite integral:

\int_0^4\left(9 e^{0.25 x}+6 x\right) d x \approx 36 (2.71828) + 12

Simplifying the expression, we get:

\int_0^4\left(9 e^{0.25 x}+6 x\right) d x \approx 97.41888 + 12

Simplifying the expression, we get:

\int_0^4\left(9 e^{0.25 x}+6 x\right) d x \approx 109.41888

Conclusion

In this article, we evaluated the given definite integral $\int_0^4\left(9 e^{0.25 x}+6 x\right) d x$ and expressed the answer in its simplest form. We used the Fundamental Theorem of Calculus to evaluate the definite integral, and we approximated the value of the definite integral using the value of $e^1$ .

The final answer is $\boxed{109.41888}$ .

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Introduction

In our previous article, we evaluated the definite integral $\int_0^4\left(9 e^{0.25 x}+6 x\right) d x$ and expressed the answer in its simplest form. In this article, we will answer some frequently asked questions related to evaluating definite integrals.

Q: What is a definite integral?

A: A definite integral is a mathematical operation that calculates the area under a curve or the accumulation of a quantity over a given interval. It is denoted by the symbol $\int_a^b f(x) dx$ , where $f(x)$ is the function being integrated, and $a$ and $b$ are the limits of integration.

Q: How do I evaluate a definite integral?

A: To evaluate a definite integral, you need to follow these steps:

Integrate the function: First, you need to integrate the function $f(x)$ with respect to $x$ .
Apply the Fundamental Theorem of Calculus: Then, you need to apply the Fundamental Theorem of Calculus to evaluate the definite integral.
Evaluate the antiderivative: Finally, you need to evaluate the antiderivative of the function at the limits of integration.

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 $a$ to $b$ is equal to the antiderivative of $f(x)$ evaluated at $b$ minus the antiderivative of $f(x)$ evaluated at $a$ . In other words, it states that:

\int_a^b f(x) dx = F(b) - F(a)

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

Q: How do I apply the Fundamental Theorem of Calculus?

A: To apply the Fundamental Theorem of Calculus, you need to follow these steps:

Find the antiderivative: First, you need to find the antiderivative of the function $f(x)$ .
Evaluate the antiderivative: Then, you need to evaluate the antiderivative at the limits of integration.
Subtract the values: Finally, you need to subtract the value of the antiderivative at the lower limit of integration from the value of the antiderivative at the upper limit of integration.

Q: What are some common mistakes to avoid when evaluating definite integrals?

A: Here are some common mistakes to avoid when evaluating definite integrals:

Not following the order of operations: Make sure to follow the order of operations when evaluating the definite integral.
Not evaluating the antiderivative correctly: Make sure to evaluate the antiderivative correctly at the limits of integration.
Not subtracting the values correctly: Make sure to subtract the values correctly when applying the Fundamental Theorem of Calculus.

Q: How do I check my work when evaluating definite integrals?

A: Here are some steps to check your work when evaluating definite integrals:

Re-evaluate the antiderivative: Re-evaluate the antiderivative of the function at the limits of integration.
Check the Fundamental Theorem of Calculus: Check that you have applied the Fundamental Theorem of Calculus correctly.
Check the order of operations: Check that you have followed the order of operations when evaluating the definite integral.

Conclusion

In this article, we answered some frequently asked questions related to evaluating definite integrals. We discussed the definition of a definite integral, how to evaluate a definite integral, and how to apply the Fundamental Theorem of Calculus. We also discussed some common mistakes to avoid and how to check your work when evaluating definite integrals.

By following these steps and avoiding common mistakes, you can evaluate definite integrals with confidence and accuracy.