Introduction to Definite Integrals: Videos & Practice Problems

To find the area under a curve, we use the concept of the definite integral, which is represented as $\int f_a{f}_{b}dx$. This integral calculates the true area by summing an infinite number of rectangles under the curve. Key rules include the sum and difference rule, constant multiple rule, and the additivity rule, which allows combining integrals over adjacent intervals. Understanding these principles is essential for accurately solving definite integrals.

Definition of the Definite Integral

Estimating the area under a curve can be approached by filling the area with rectangles and calculating their combined area, a method known as Riemann sums. However, to achieve a precise measurement of the area under a function, we can utilize the concept of the definite integral. This method involves taking the limit as the number of rectangles approaches infinity, which allows us to fill the area under the curve more accurately.

The definite integral is represented mathematically as:

$$\int_{a}^{b} f(x) \, dx$$

In this expression, \(a\) and \(b\) are the lower and upper bounds on the x-axis, respectively, and \(f(x)\) is the function being integrated. The integral sign indicates that we are summing the areas of infinitely many rectangles under the curve between these two bounds.

To illustrate how to set up a definite integral, consider an example where we need to express a limit as a definite integral over the interval from 0 to 4. We start by identifying the function and recognizing that the summation resembles a Riemann sum, with the number of rectangles approaching infinity. This leads us to set up the definite integral from 0 to 4:

$$\int_{0}^{4} (x + 1) \, dx$$

Next, to find the exact area represented by this integral, we can analyze the graph of the function to identify familiar geometric shapes. In this case, the area can be divided into a triangle and a rectangle. The area of the triangle can be calculated using the formula:

$$\text{Area}_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height}$$

Here, the base is 4 (from 0 to 4 on the x-axis) and the height is 4 (from 1 to 5 on the y-axis), resulting in:

$$\text{Area}_{\text{triangle}} = \frac{1}{2} \times 4 \times 4 = 8$$

For the rectangle, the area is calculated as:

$$\text{Area}_{\text{rectangle}} = \text{base} \times \text{height}$$

With a base of 4 and a height of 1, we find:

$$\text{Area}_{\text{rectangle}} = 4 \times 1 = 4$$

Adding the areas of the triangle and rectangle gives the total area under the curve:

$$\text{Total Area} = 8 + 4 = 12$$

This process demonstrates how to set up definite integrals and calculate the exact area under a curve, highlighting the importance of understanding both the integral setup and the geometric interpretation of the area. As you continue to explore definite integrals, you will encounter various applications and rules that will deepen your understanding of this fundamental concept in calculus.

Find $\int_0^6\!f\left(x\right)\,dx$ given the graph of $y=f\left(x\right)$.

• A. 9
• B. 11
• C. 18
• D. 3

Answer: (B)

Express the following limit as a definite integral on the interval $[0,10]$.
$\lim_{n\to\infty}\sum_{k=1}^{n}\left(x_{k}^{\ast}-3\right)^2\Delta x$

• A. $\int_{10}^0\!\left(x-3\right)^2\,dx$
• B. $\int_0^{10}\!\left(x-3\right)^2\,dx$
• C. $\int_0^{\infty}\!\left(x-3\right)^2\,dx$
• D. $\int_{\infty}^0\!\left(x-3\right)^2\,dx$

Answer: (B)

Basic Rules for Definite Integrals

Definite integrals are a fundamental concept in calculus, representing the area under a curve between two specified bounds. Understanding the basic rules that apply to definite integrals is essential for solving problems effectively. Many of these rules mirror those of indefinite integrals, making them easier to grasp.

One of the primary rules is the sum and difference rule, which states that the integral of a sum or difference of functions can be separated into the sum or difference of their individual integrals. For example, if you have the integral of \( f(x) + g(x) \) from \( a \) to \( b \), it can be expressed as:

\[\int_a^b (f(x) + g(x)) \, dx = \int_a^b f(x) \, dx + \int_a^b g(x) \, dx\]

Here, the bounds \( a \) and \( b \) remain unchanged for both integrals.

Another important rule is the constant multiple rule, which allows you to factor out constants from the integral. If you have a constant \( k \) multiplied by a function \( f(x) \), the integral can be simplified as follows:

\[\int_a^b k \cdot f(x) \, dx = k \cdot \int_a^b f(x) \, dx\]

This rule also preserves the bounds of integration.

Moving on to more specific rules for definite integrals, the order of integration rule states that if you switch the bounds of integration, the integral's value becomes negative. For instance, switching the bounds from \( b \) to \( a \) results in:

\[\int_b^a f(x) \, dx = -\int_a^b f(x) \, dx\]

This is crucial when dealing with integrals where the upper limit is less than the lower limit.

The zero width interval rule indicates that if the upper and lower bounds are the same (i.e., \( a \) to \( a \)), the integral evaluates to zero:

\[\int_a^a f(x) \, dx = 0\]

This is because there is no area under the curve when the bounds do not span any interval.

Lastly, the additivity rule allows you to combine integrals over adjacent intervals. If you have two integrals from \( a \) to \( c \) and from \( c \) to \( b \), where \( c \) is between \( a \) and \( b \), you can express this as a single integral from \( a \) to \( b \):

\[\int_a^c f(x) \, dx + \int_c^b f(x) \, dx = \int_a^b f(x) \, dx\]

These rules provide a solid foundation for working with definite integrals, and applying them in various combinations will help solve more complex problems. Practice with these concepts will enhance your understanding and proficiency in calculus.

Given the following definite integral of the function $f\left(x\right)=3x^2-2x$, write the simplified integral:
-$\int_4^0\!f\left(x\right)\,dx$

• A. $2\int_4^0\!x\,dx-3\int_4^0\!x^2\,dx$
• B. $3\int_4^0\!x^2\,dx-2\int_4^0\!x\,dx$
• C. $2\int_0^4\!x\,dx-3\int_0^4\!x^2\,dx$
• D. $3\int_0^4\!x^2\,dx-2\int_0^4\!x\,dx$

Answer: (D)

Write the two definite integrals subtracted below as a single integral.
$\int_1^6\!\sqrt{x^2-5x}\,dx-\int_{10}^6\!\sqrt{x^2-5x}\,dx$

• A. $\int_{10}^1\!\sqrt{x^2-5x}\,dx$
• B. $\int_1^{10}\!\sqrt{x^2-5x}\,dx$
• C. $\int_6^9\!\sqrt{x^2-5x}\,dx$
• D. $\int_9^6\!\sqrt{x^2-5x}\,dx$

Answer: (B)

Here’s what students ask on this topic:

What is the difference between definite and indefinite integrals?

Definite and indefinite integrals are both fundamental concepts in calculus, but they serve different purposes. A definite integral, denoted as $\int a_{b}f\left(x\right)dx$, calculates the exact area under a curve between two specified bounds, $a$ and $b$. It results in a numerical value representing this area. In contrast, an indefinite integral, represented as $\int f\left(x\right)dx$, finds the antiderivative of a function, resulting in a general function plus a constant of integration, $C$. This represents a family of functions. While definite integrals provide a specific area, indefinite integrals help in understanding the accumulation of quantities.

How do you calculate the area under a curve using definite integrals?

To calculate the area under a curve using definite integrals, you first need to identify the function $f\left(x\right)$ and the interval $[a,b]$ over which you want to find the area. The definite integral is expressed as $\int a_{b}f\left(x\right)dx$. This integral represents the limit of Riemann sums as the number of rectangles approaches infinity, providing the exact area under the curve. You can evaluate this integral using fundamental integration techniques or numerical methods if the function is complex. The result is a numerical value that represents the total area between the curve and the x-axis over the specified interval.

What are the basic rules for definite integrals?

Definite integrals have several basic rules that simplify calculations. The sum and difference rule allows you to split the integral of a sum or difference of functions into separate integrals. The constant multiple rule lets you factor out constants from the integral. The order of integration rule states that reversing the limits of integration changes the sign of the integral. The zero width interval rule indicates that integrating over an interval with the same bounds results in zero. Lastly, the additivity rule allows you to combine integrals over adjacent intervals into a single integral. These rules help in efficiently solving integrals and understanding their properties.

How does the order of integration rule affect definite integrals?

The order of integration rule for definite integrals states that if you switch the limits of integration, the sign of the integral changes. For example, if you have an integral from $a$ to $b$, switching it to go from $b$ to $a$ will result in the negative of the original integral. Mathematically, this is expressed as $\int a_{b}f\left(x\right)dx$ = -$\int b_{a}f\left(x\right)dx$. This rule is crucial when evaluating integrals, as it ensures the correct orientation of the area being calculated.

What is the additivity rule for definite integrals?

The additivity rule for definite integrals states that if you have two integrals over adjacent intervals, you can combine them into a single integral. Specifically, if you have an integral from $a$ to $c$ and another from $c$ to $b$ for the same function, you can write it as a single integral from $a$ to $b$. Mathematically, this is expressed as $\int a_{c}f\left(x\right)dx$ + $\int c_{b}f\left(x\right)dx$ = $\int a_{b}f\left(x\right)dx$. This rule is useful for breaking down complex integrals into simpler parts or combining simpler integrals into a comprehensive solution.