Evaluating definite integrals

The Evaluating definite integrals exercise appears under the Integral calculus Math Mission. This exercise uses the Fundamental Theorem of Calculus to find the value of a definite integral.

Types of Problems

There are two types of problems in this exercise:

1. Find the value of the antiderivative function: This problem presents a function defined in terms of an integral. The student is expected to find the antiderivative of the function to get a general solution, use a given initial condition to find a particular solution, then use the particular solution to answer a question.

   

   Find the value of the antiderivative function

2. Evaluate the definite integral: This problem provides a definite integral. The student is expected to find the exact value and write it in the space provided.

   

   Evaluate the definite integral

Strategies

Knowledge of the fundamental theorem of calculus and some basic antiderivatives are encouraged to ensure success on this exercise.

1. For polynomial pieces, the antiderivative of $a x^n$ is $\frac{(ax^{(n+1)})}{(n+1)}$.
2. The value a definite integral is the difference between the antiderivative evaluated at the upper and lower bounds.
3. On the first problem type, one should use the initial condition to find the value of the constant of integration.
4. It is possible to avoid finding the constant of integration if one uses the FTC to realize that $f(b)=f(a)+\int_a^bf(t)dt$.

Real-life Applications

1. Integrals can be used in physics to find the total distance travelled by objects.
2. Integrals can be used in economics to find the total profit or cost of business ventures.
3. Many times situations where one is looking for a "total" can be performed with integration.