Integration is a mathematical concept that involves finding the area under a curve or the accumulation of a quantity over an interval. It is important because it allows us to solve real-world problems involving rates of change, such as calculating the distance traveled by an object with a changing velocity or finding the total cost of a varying rate of production.
Indefinite integration involves finding the most general antiderivative of a function, while definite integration involves finding the specific value of the area under the curve between two given points. In other words, indefinite integration gives a function while definite integration gives a number.
To evaluate a definite integral, you first need to find the indefinite integral of the function. Then, you substitute the upper and lower limits of integration into the indefinite integral and subtract the resulting values to find the definite integral.
Some common methods for evaluating definite integrals include the Fundamental Theorem of Calculus, substitution, integration by parts, and trigonometric substitution. Which method to use depends on the complexity of the function and your familiarity with the different techniques.
You can check your answer by taking the derivative of the definite integral. If the derivative matches the original function, then your answer is correct. You can also use online calculators or graphing tools to visualize the area under the curve and compare it to your calculated answer.