Introduction to Definite Integrals: Videos & Practice Problems

To find the area under a curve, we use the concept of the definite integral, which represents the true area as the limit of Riemann sums with an infinite number of rectangles. The definite integral is expressed as ${\int }_a^{b}f\left(x\right)dx$. Key rules include the sum and difference rule, constant multiple rule, order of integration rule, zero width interval rule, and additivity rule, which help simplify calculations and understand the properties of integrals.

Definition of the Definite Integral

Estimating the area under a curve can be approached by using rectangles to approximate the area, a method known as Riemann sums. However, to achieve a precise calculation of the area, we can utilize the concept of the definite integral. The definite integral allows us to find the exact area under a curve by considering an infinite number of rectangles, which leads to a more accurate representation of the area.

The definite integral is expressed mathematically as:

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

In this notation, \(a\) and \(b\) represent the lower and upper bounds on the x-axis, respectively, while \(f(x)\) is the function whose area we are calculating. The integral sign indicates that we are summing up the areas of infinitely many rectangles under the curve from \(x = a\) to \(x = b\).

To set up a definite integral, we first identify the function and the interval over which we want to calculate the area. For example, if we are given a function and asked to express a limit as a definite integral from 0 to 4, we would write:

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

Next, to find the exact area represented by this integral, we can break the area under the curve into familiar geometric shapes. For instance, if the area consists of a triangle and a rectangle, we can calculate their areas separately. The area of a triangle is given by:

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

In our example, if the base of the triangle is 4 and the height is 4, the area would be:

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

For the rectangle, the area is calculated using the formula:

\[\text{Area}_{\text{rectangle}} = \text{base} \times \text{height}\]

If the base is 4 and the height is 1, the area would be:

\[\text{Area}_{\text{rectangle}} = 4 \times 1 = 4\]

By adding the areas of the triangle and rectangle together, we find the total area under the curve:

\[\text{Total Area} = 8 + 4 = 12\]

This process illustrates how to set up and evaluate definite integrals to find the true area under a curve. Understanding this concept is crucial as it forms the foundation for more advanced applications of calculus, including various rules and techniques for solving definite integrals.

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 $[0,10]$.
${\lim }_{n\to \infty }{\sum }_{k=1}^n{(x_k^{\ast }-3)}^2\Delta x$

• A. ${\int }_{10}^0{(x-3)}^{2}dx$
• B. ${\int }_0^{10}{(x-3)}^{2}dx$
• C. ${\int }_0^{\infty }{(x-3)}^{2}dx$
• D. ${\int }_{\infty }^0{(x-3)}^{2}dx$

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:

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

The bounds \( a \) and \( b \) remain unchanged for each integral.

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:

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

Again, the bounds \( a \) and \( b \) remain the same.

Moving on to more specific rules for definite integrals, the order of integration rule states that if you switch the bounds of the integral, the result will be the negative of the original integral. For instance, switching the bounds from \( b \) to \( a \) gives:

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

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 interval has no width.

Lastly, the additivity rule allows you to combine two integrals that share a common bound. If you have an integral from \( a \) to \( c \) and another from \( c \) to \( b \), you can combine them into a single integral from \( a \) to \( b \):

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

This rule is particularly useful for simplifying calculations when dealing with piecewise functions or functions defined over multiple intervals.

By applying these rules, you can tackle a variety of problems involving definite integrals. Practice with examples will help solidify your understanding and ability to use these rules effectively in more complex scenarios.

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. $2{\int }_4^{0}xdx-3{\int }_4^0{x}^{2}dx$
• B. $3{\int }_4^0{x}^{2}dx-2{\int }_4^{0}xdx$
• C. $2{\int }_0^{4}xdx-3{\int }_0^4{x}^{2}dx$
• D. $3{\int }_0^4{x}^{2}dx-2{\int }_0^{4}xdx$

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)

Evaluate the following definite integral.
${\int }_{100}^{100}\frac{\sin \left(2x\right)-100x^2}{\sqrt{x^5+\tan (3x-6)}}dx$

• A. 1
• B. Indeterminate
• C. 0

Answer: (C)

Here’s what students ask on this topic:

What is the difference between definite and indefinite integrals?

Definite integrals and indefinite integrals are two fundamental concepts in calculus. A definite integral, denoted as $\int a_{b}f\left(x\right)dx$, calculates the exact area under a curve between two specific bounds, a and b, on the x-axis. It results in a numerical value representing this area. In contrast, an indefinite integral, denoted as $\int f\left(x\right)dx$, represents a family of functions and includes a constant of integration, C. It does not have specific bounds and is used to find antiderivatives. Understanding these differences is crucial for solving calculus problems effectively.

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

To calculate the area under a curve using definite integrals, you first identify the function f(x) representing the curve 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 calculates the limit of Riemann sums as the number of rectangles approaches infinity, providing the exact area. You can solve the integral using integration techniques, such as substitution or integration by parts, and apply the fundamental theorem of calculus to evaluate it at the bounds a and b. The result is the area under the curve between these bounds.

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 switching the bounds of integration changes the sign of the integral. The zero width interval rule indicates that integrating over the same bounds results in zero. Lastly, the additivity rule allows you to combine integrals over adjacent intervals into a single integral. These rules are essential for efficiently solving definite integrals and understanding their properties.

How does the limit of Riemann sums relate to definite integrals?

The limit of Riemann sums is foundational to understanding definite integrals. Riemann sums approximate the area under a curve by dividing the interval [a, b] into subintervals and summing the areas of rectangles formed under the curve. As the number of rectangles increases, the approximation becomes more accurate. The definite integral is the limit of these Riemann sums as the number of rectangles approaches infinity, providing the exact area under the curve. This concept is expressed mathematically as $\int a_{b}f\left(x\right)dx$, representing the true area between the bounds a and b.