Evaluating the First Definite Integral

Oct 13, 2025 by magicalceylontours.com 94 views

Introduction

In calculus, definite integrals play a crucial role in understanding the behavior of functions and solving problems in various fields. A definite integral is a mathematical operation that calculates the area under a curve or the accumulation of a quantity over a specified interval. In this article, we will evaluate three definite integrals using the fundamental theorem of calculus.

Evaluating the First Definite Integral

1. $\int_1^2 (2x-1) \, dx$

To evaluate this definite integral, we will use the power rule of integration, which states that $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ . We will also use the constant multiple rule, which states that $\int af(x) \, dx = a\int f(x) \, dx$ .

# Step 1: Integrate the function
∫(2x-1) dx = ∫2x dx - ∫1 dx
= 2∫x dx - ∫1 dx
= 2 * (x^2/2) - x + C
= x^2 - x + C

2: Apply the Fundamental Theorem of Calculus

The fundamental theorem of calculus states that $\int_a^b f(x) \, dx = F(b) - F(a)$ , where $F(x)$ is the antiderivative of $f(x)$ . In this case, we have:

# Step 2: Evaluate the antiderivative at the limits of integration
F(2) = (2)^2 - (2) = 4 - 2 = 2
F(1) = (1)^2 - (1) = 1 - 1 = 0

3: Calculate the definite integral

Now, we can calculate the definite integral by subtracting the value of the antiderivative at the lower limit from the value of the antiderivative at the upper limit:

# Step 3: Calculate the definite integral
∫[1,2] (2x-1) dx = F(2) - F(1) = 2 - 0 = 2

Evaluating the Second Definite Integral

1. $\int_0^{-1} x^3 \, dx$

# Step 1: Integrate the function
∫x^3 dx = (x^4)/4 + C

2: Apply the Fundamental Theorem of Calculus

The fundamental theorem of calculus states that $\int_a^b f(x) \, dx = F(b) - F(a)$ , where $F(x)$ is the antiderivative of $f(x)$ . In this case, we have:

# Step 2: Evaluate the antiderivative at the limits of integration
F(-1) = ((-1)^4)/4 = 1/4
F(0) = (0^4)/4 = 0

3: Calculate the definite integral

Now, we can calculate the definite integral by subtracting the value of the antiderivative at the lower limit from the value of the antiderivative at the upper limit:

# Step 3: Calculate the definite integral
∫[0,-1] x^3 dx = F(-1) - F(0) = 1/4 - 0 = 1/4

Evaluating the Third Definite Integral

1. $\int_2^3 (x-1) \, dx$

# Step 1: Integrate the function
∫(x-1) dx = ∫x dx - ∫1 dx
= x^2/2 - x + C

2: Apply the Fundamental Theorem of Calculus

The fundamental theorem of calculus states that $\int_a^b f(x) \, dx = F(b) - F(a)$ , where $F(x)$ is the antiderivative of $f(x)$ . In this case, we have:

# Step 2: Evaluate the antiderivative at the limits of integration
F(3) = (3^2)/2 - (3) = 9/2 - 3 = 3/2
F(2) = (2^2)/2 - (2) = 4/2 - 2 = 0

3: Calculate the definite integral

Now, we can calculate the definite integral by subtracting the value of the antiderivative at the lower limit from the value of the antiderivative at the upper limit:

# Step 3: Calculate the definite integral
∫[2,3] (x-1) dx = F(3) - F(2) = 3/2 - 0 = 3/2

Conclusion

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 specified interval. It is denoted by the symbol ∫ and is used to find the area between a curve and the x-axis.

Q: How do I evaluate a definite integral?

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

Integrate the function using the power rule of integration and the constant multiple rule.
Apply the fundamental theorem of calculus to calculate the definite integral.
Evaluate the antiderivative at the upper and lower limits of integration.
Subtract the value of the antiderivative at the lower limit from the value of the antiderivative at the upper limit.

Q: What is the fundamental theorem of calculus?

A: The fundamental theorem of calculus states that ∫[a,b] f(x) dx = F(b) - F(a), where F(x) is the antiderivative of f(x). This theorem allows us to calculate definite integrals by evaluating the antiderivative at the upper and lower limits of integration.

Q: How do I use the power rule of integration?

A: The power rule of integration states that ∫x^n dx = (x^(n+1))/(n+1) + C. To use this rule, you need to identify the power of x in the function and then apply the rule.

Q: How do I use the constant multiple rule of integration?

A: The constant multiple rule of integration states that ∫af(x) dx = a∫f(x) dx. To use this rule, you need to identify the constant multiple of the function and then apply the rule.

Q: What is the difference between a definite integral and an indefinite integral?

A: A definite integral is a mathematical operation that calculates the area under a curve or the accumulation of a quantity over a specified interval. An indefinite integral, on the other hand, is a function that represents the antiderivative of a given function. The definite integral is used to find the area between a curve and the x-axis, while the indefinite integral is used to find the antiderivative of a function.

Q: How do I apply the fundamental theorem of calculus to calculate a definite integral?

A: To apply the fundamental theorem of calculus, you need to follow these steps:

Evaluate the antiderivative at the upper limit of integration.
Evaluate the antiderivative at the lower limit of integration.
Subtract the value of the antiderivative at the lower limit from the value of the antiderivative at the upper limit.

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

A: Some common mistakes to avoid when evaluating definite integrals include:

Forgetting to apply the fundamental theorem of calculus.
Forgetting to evaluate the antiderivative at the upper and lower limits of integration.
Forgetting to subtract the value of the antiderivative at the lower limit from the value of the antiderivative at the upper limit.
Not using the correct rules of integration, such as the power rule and the constant multiple rule.

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

A: To check your work when evaluating a definite integral, you need to follow these steps:

Evaluate the definite integral using the fundamental theorem of calculus.
Check that the antiderivative is evaluated at the correct limits of integration.
Check that the value of the antiderivative at the upper limit is subtracted from the value of the antiderivative at the lower limit.
Check that the result is a numerical value, not a function.