Definite Integral Game

Definite Integral Game
This game focuses on evaluating definite integrals using the Fundamental Theorem of Calculus. It includes questions ranging from simple polynomials to trigonometric and exponential functions with specific limits. Select one of the four possible answers. If you are right, the answer will be highlighted in green. If you are wrong, the wrong answer will be highlighted in red and correct answer will be highlighted in green to help you master this skill. Scroll down the page for a more detailed explanation.

 

 

How to Play the Definite Integral Game
The game will show you a definite integral. Your task is to evaluate the integral and select the correct answer.
Here’s how to play:

1. Start: Each Quiz consists of 10 questions.
   
2. Look at the Problem: You will be given a Integral.
   
3. Select Your Answer: Evaluate and select your answer.br>
4. Check Your Work: If you are right, the answer will be highlighted in green. If you are wrong, the wrong answer will be highlighted in red and correct answer will be highlighted in green. The game will also give you a hint.
   
5. Get a New Problem: Click “Next Integral” for a new problem.
   Your score is tracked, showing how many you’ve gotten right.
   
6. Finish Game When you have completed 10 questions, click “Finish Game” to get your final score.
    
   

Evaluating the Definite Integral
The definite integral is used to find the net signed area between a function \(f(x)\) and the \(x\)-axis over a specific interval \([a, b]\).
Definition and Notation
The Riemann Sum (Conceptual Definition)
The definite integral is conceptually defined as the limit of a Riemann Sum. This means we are approximating the area under the curve by summing the area of many infinitesimally thin rectangles, then taking the limit as the width of those rectangles approaches zero.
\(\int_a^b f(x) \, dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) \Delta x\)

\(\int\) is the integral sign, which represents summation (an elongated ‘S’).
\(a\) is the lower limit of integration.
\(b\) is the upper limit of integration.
\(f(x)\) is the integrand (the function).
\(dx\) indicates the variable of integration.
The result, \(\int_a^b f(x) \, dx\), is a single number, representing the area.
 

The Fundamental Theorem of Calculus (FTC), Part 2
Fortunately, we don’t have to use the cumbersome limit definition every time. The Fundamental Theorem of Calculus (Part 2) provides a powerful shortcut, linking anti-differentiation to the definite integral.
Theorem Statement
If \(f\) is continuous on the interval \([a, b]\), and \(F(x)\) is any antiderivative of \(f(x)\) (meaning \(F’(x) = f(x)\)), then: \(\int_a^b f(x) \, dx = F(b) - F(a)\)
This theorem states that the definite integral of a function over an interval is found by evaluating its antiderivative at the upper limit (\(b\)) and subtracting its antiderivative evaluated at the lower limit (\(a\)).

Step-by-Step Evaluation Process
Here is the standard procedure for evaluating any definite integral:
Step 1: Find the Antiderivative
Ignore the limits of integration (\(a\) and \(b\)) temporarily and find the indefinite integral (the general antiderivative, \(F(x)\)) of \(f(x)\).
Crucial Note: When finding \(F(x)\), you do not need to include the constant of integration (\(+C\)), because it would cancel out in the final subtraction: \((F(b) + C) - (F(a) + C) = F(b) - F(a)\)
Step 2: Use the Evaluation Notation
Write the antiderivative \(F(x)\) enclosed in square brackets, with the limits \(a\) and \(b\) placed as a subscript and superscript on the right side.
\(\int_a^b f(x) \, dx = [F(x)]_a^b\)
Step 3: Apply the Fundamental Theorem
Substitute the upper limit (\(b\)) into \(F(x)\), and then subtract the result of substituting the lower limit (\(a\)) into \(F(x)\).
\([F(x)]_a^b = F(b) - F(a)\)

Example
Evaluate the definite integral \(\int_1^3 (x^2 + 2) , dx\).
Step 1: Find the Antiderivative
The antiderivative of \(f(x) = x^2 + 2\) is \(F(x) = \frac{x^3}{3} + 2x\).
Step 2: Apply Evaluation Notation
\(\int_1^3 (x^2 + 2) \, dx = \left[ \frac{x^3}{3} + 2x \right]_1^3\)
Step 3: Apply the Fundamental Theorem
\(F(3) - F(1)\)
Evaluate at the Upper Limit (\(b=3\)):
\(F(3) = \frac{(3)^3}{3} + 2(3) = \frac{27}{3} + 6 = 9 + 6 = 15\)
Evaluate at the Lower Limit (\(a=1\)):
\(F(1) = \frac{(1)^3}{3} + 2(1) = \frac{1}{3} + 2 = \frac{1}{3} + \frac{6}{3} = \frac{7}{3}\)
Subtract:
\(F(3) - F(1) = 15 - \frac{7}{3} = \frac{45}{3} - \frac{7}{3} = \frac{38}{3}\)
Therefore, \(\int_1^3 (x^2 + 2) \, dx = \frac{38}{3}\).

This video gives a clear, step-by-step approach to learn how to find Definite Integrals.

• Show Video

 

Derivative Games
Derivative Challenge (Polynomials)
Find the derivatives of polynomials using the Power, Sum, Difference, and Constant Multiple rules.

Derivative Challenge (Binomials)
Find the derivatives of the product two binomial functions. Either use the Product Rule or multiply the two binomials first to get a single polynomial and use the Power Rule.

Derivative Challenge (Product Rule)
Find the derivatives of the product a polynomial and a trig function. Use the Product Rule.

Derivative Challenge (Quotient Rule)
Find the derivatives of a function using the Quotient Rule.

Derivative Challenge (Chain Rule)
Find the derivatives of a function using the Chain Rule.

Derivative Rules
Find the derivatives of a function using the Power Rule, Constant Rule, Sum/Difference Rule, Product Rule, Quotient Rule, Chain Rule, and the rules for trigonometric and exponential functions.

Derivative Challenge (Trig Function)
Find the derivatives of Trigonometric Functions. May require other basic rules for example the Chain Rule, Product Rule etc.

Derivative Challenge (Exp, Log)
Find the derivatives of Exponential and Logarithmic Functions. May require other basic rules for example the Chain Rule, Product Rule etc.

Derivative Challenge (Hyperbolic)
Find the derivatives of Hyperbolic Functions. May require other basic rules for example the Chain Rule, Product Rule etc.

Integral Games
Power Rule Practice
Practice using the Power Rule for Integration.

Exponential Integration Station
Practice using the Exponential Rule for Integration.

Definite Integral Quest
Practice evaluating Trig Integrals.
 

Try out our new and fun Fraction Concoction Game.

Add and subtract fractions to make exciting fraction concoctions following a recipe. There are four levels of difficulty: Easy, medium, hard and insane. Practice the basics of fraction addition and subtraction or challenge yourself with the insane level.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.