Limits of integration are used in a type of integration called definite integration. When we add limits to a normal (indefinite) integral, it becomes a definite integral, which gives us a specific value.

For example, in the expression:
∫ from b to a of f(x) dx,

• f(x) is the function we want to integrate.
• a is the upper limit.
• b is the lower limit.

To solve this, we follow two simple steps:

1. Integrate the function to get its antiderivative (let’s call it F(x)).
2. Apply the limits by subtracting the value of F(x) at the lower limit from the value at the upper limit.
   That is: ∫ from b to a of f(x) dx = F(a) − F(b)

This process helps us calculate the area under a curve between two points on the x-axis.

In this article, you’ll learn how to solve definite integrals using limits. We’ll look at useful formulas, step-by-step methods, simple examples, and common questions to help you understand better.

What are the Limits of Integration?

Let’s say we are given a function f(x). To find the area under the curve between two points on the x-axis, we use a method called definite integration.

A definite integral is a way to calculate the total area between a curve and the x-axis from one point to another. These two points are called the limits of integration.

• The Lower Limit (usually written as a) is the starting point of the interval.
• The Upper Limit (usually written as b) is the ending point of the interval.

To solve a definite integral, we follow two steps:

1. First, we find the antiderivative of the function. This means finding another function, called F(x), whose derivative is f(x).
2. Next, we apply the limits. We plug in the upper and lower limits into F(x), and subtract:
   F(b) − F(a)

This result gives us the area under the curve between x = a and x = b. This method is useful in many real-life situations, like finding distance, work done, or total quantity from a rate function.

The formula looks like this:

\(\int_{b}^{a}f(x).dx = [F(x)]_{b}^{a} = F(a) - F(b)\)

These limits (a and b) help us set the range of x-values we are looking at. The result is just a number—it tells us how much "space" is under the curve between those two points.

Remember:

• This is called a definite integral.
• It gives a specific number as an answer.
• Unlike indefinite integrals, there is no "+C" constant at the end.

The limits of integration involve two steps. First, we solve the integration problem. Then, we apply the limits to find two values of the function. The final value of the limits of integration is obtained by taking the difference between these two values.

Step 1: To find the integral of a function f(x) between the limits [a, b], we need to calculate its antiderivative. It is represented as \(\int_{b}^{a}f(x).dx = [F(x)]_{b}^{a}\).

Step 2: Once we have the antiderivative, we apply the limits \([a, b]\) to it by substituting the values of \(a\) and \(b\) into the antiderivative. This gives us the final answer, which is obtained by subtracting \(F(a)\) from \(F(b)\). This can be represented as \([F(x)]\) evaluated from \(a\) to \(b\) \(= F(a) - F(b)\).

Therefore, the limits of integration help us determine the exact numerical value of the given integral expression.

How to Change the Limits of Integration?

1. Find the Original Limits:
   First, check the current limits of the integral. These are usually written as "from a to b".
   Here, a is the lower limit and b is the upper limit.
2. Choose New Limits:
   Decide what you want the new limits to be. Call them c (new lower limit) and d (new upper limit).
3. Adjust the Limits:
   If you're making a change of variable (like substitution), you'll also need to update the limits according to the new variable.
   • For example, if you make a substitution like x = u + 2, then you'll plug the original x values into this equation to find the new u values.
4. Change the Variable in the Function:
   Don’t forget to rewrite the function in terms of the new variable (if you've made a substitution). Also update dx accordingly (for example, if x = u + 2, then dx = du).
5. Use the New Limits:
   Once everything is changed (the function and the limits), you can solve the integral using the new form and new limits.

Formulas of Limits of Integration

The formulas we'll discuss next are crucial when working with definite integrals. They help us solve integration problems by using the lower and upper limits to find the final answer of the integral. These formulas make it easier to integrate the given function and obtain its specific value.

• \(\int_{a}^{b}f(x)dx = \int_{a}^{b}f(t)dt\)
• \(\int_{a}^{b}f(x)dx = -\int_{b}^{a}f(x)dx\)
• \(\int_{a}^{b}cf(x)dx = c\int_{a}^{b}f(x)dx\)
• \(\int_{a}^{b}(f(x) \pm g(x))dx = \int_{a}^{b}f(x)dx \pm \int_{a}^{b}g(x)dx\)
• \(\int_{a}^{b}f(x)dx = \int_{a}^{c}f(x)dx + \int_{c}^{b}f(x)dx\)
• \(\int_{a}^{b}f(x)dx = \int_{a}^{b}f(a+b-x)dx\)
• \(\int_{0}^{a}f(x)dx = \int_{0}^{a}f(a-x)dx\)
• \(\int_{0}^{2a}f(x)dx = 2\int_{0}^{a}f(x)dx\), if \(f(2a-x) = f(x)\)
• \(\int_{0}^{2a}f(x)dx = 0\), if \(f(2a-x) = -f(x)\)
• \(\int_{-a}^{a}f(x).dx = 2\int_{0}^{a}f(x)dx\), if \(f(x)\) is an even function, and \(f(-x) = f(x)\)
• \(\int_{-a}^{a}f(x).dx = 0\), if \(f(x)\) is an odd function, and \(f(-x) = -f(x)\)

To learn about even and odd functions in detail.

Application of Limits of Integration

Let's explore the diverse practical applications of limits of integration:

• Physics: Limits of integration are used to calculate displacement, velocity, and acceleration of objects. By integrating velocity functions, we can determine an object's position or the distance it has travelled.
• Economics: Integration plays a crucial role in analyzing production and cost functions. Limits of integration help us examine the behaviour of variables as quantities approach extremes, enabling us to make informed decisions.
• Engineering: Limits of integration find applications in various engineering fields. In fluid mechanics, they are used to solve problems involving fluid flow and pressure distribution. In electrical circuits, limits help analyze signals and determine system behaviour.
• Probability and Statistics: Limits of integration are employed in calculating probabilities and finding areas under probability density functions. They help us understand the likelihood of events occurring within a given range.
• Optimization Problems: Limits of integration are crucial in solving optimization problems, where we seek to find the maximum or minimum values of a function. By applying appropriate limits, we can determine the optimal solution.

Limits of Integration Solved Examples

1) Calculate the integral by utilizing the limits of integration for the expression \(\int_{-3}^{3}x^{4}dx\).

Solution:

\(\int_{-3}^{3}x^{4}dx = 2\int_{0}^{3}x^{4}dx\)

\(\Rightarrow\) \(\int_{-3}^{3}x^{4}dx = 2\left[\frac{x^{5}}{5}\right]_{0}^{3}\)

\(\Rightarrow\) \(\int_{-3}^{3}x^{4}dx = 2\left[\frac{(3)^5}{5} - 0\right]\)

\(\Rightarrow\) \(\int_{-3}^{3}x^{4}dx = 2\left[\frac{243}{5}\right]\)

\(\Rightarrow\) \(\int_{-3}^{3}x^{4}dx = \frac{486}{5}\)

Therefore, the answer is \(\int_{-3}^{3}x^{4}dx = \frac{486}{5}\).

2. You have the function f(x) = 3x − 2, and you want to find the area under the curve between the points x = 2 and x = 5.
   Write and solve the definite integral for this case.

Solution:

We need to solve the integral:
I = ∫ from 2 to 5 of (3x − 2) dx

Step 1: Find the antiderivative of (3x − 2)
The antiderivative of 3x is (3x²)/2, and the antiderivative of −2 is −2x.
So, the full antiderivative is:
F(x) = (3x²)/2 − 2x

Step 2: Apply the limits of integration [2, 5]
Now, plug in the upper and lower limits:

I = [(3×5²)/2 − 2×5] − [(3×2²)/2 − 2×2]
= [(3×25)/2 − 10] − [(3×4)/2 − 4]
= [75/2 − 10] − [12/2 − 4]
= [75/2 − 10] − [6 − 4]
= (75/2 − 10) − 2
= (75/2 − 12)

Now convert 12 to a fraction:
75/2 − 12 = 75/2 − 24/2 = (75 − 24)/2 = 51/2

Answer: The area under the curve of 3x − 2 from x = 2 to x = 5 is 51/2 or 25.5 square units.

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