Problem 69 - Calculus for Biology and Medicine

Question

Find \(\int_{2}^{2} \cos \left(3 x^{2}\right) d x\).

Step-by-Step Solution

Answer

The integral evaluates to 0 because the upper and lower integration limits are the same.

Step 1: Understand the Integral

We need to evaluate the definite integral \( \int_{2}^{2} \cos(3x^2) \, dx \). This is a situation where the upper and lower limits of integration are the same (both are 2).

Step 2: Utilize the Property of Definite Integrals

A fundamental property of definite integrals is that if the upper limit and lower limit of an integral are the same, the integral evaluates to zero. This can be expressed as: \[ \int_{a}^{a} f(x) \, dx = 0. \]

Step 3: Apply the Property to the Given Integral

Apply the property from Step 2 to our integral: \[ \int_{2}^{2} \cos(3x^2) \, dx = 0. \] Since the limits of integration are identical, the value of the integral is zero.

Key Concepts

Properties of Definite Integrals

The properties of definite integrals offer handy shortcuts and simplifications for evaluating integral expressions. One vital property worth noting is when the upper limit and lower limit of an integral are the same. In this case, the definite integral always evaluates to zero. This might seem a bit surprising at first, but there's a clear reason why this property holds. When you integrate from a point to itself, no horizontal distance is covered, so the area under the curve between these points is non-existent. That's why:

• If the limits of integration are equal, the definite integral equals zero: \[ \int_{a}^{a} f(x) \, dx = 0. \]

This property can save lots of effort and prevent unnecessary calculations. It's a great example of how powerful understanding mathematical properties can be in simplifying complex-looking problems.

Integral Evaluation

Integral evaluation involves determining the value of a definite or indefinite integral. When you're asked to evaluate an integral, it means finding the exact number that represents the area under a curve for a given function. In our specific example, evaluating \( \int_{2}^{2} \cos(3x^2) \, dx \) illustrates integral evaluation using the properties of definite integrals.

• This is a definite integral with both limits the same, so applying the property: \[ \int_{2}^{2} \cos(3x^2) \, dx = 0. \]

There were no actual calculations needed because the property allowed us to directly state the result. This is an example where understanding integral properties makes evaluation straightforward and quick.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration, two central operations in calculus. This theorem states that differentiation and integration are inverse processes. It ensures that for every continuous function's integral over an interval, you can differentiate to get back to the original function.

• The theorem is comprised of two main parts:
  • The First Part connects the process of integration to antiderivatives. It states that if a function is integrable and \( F(x) \) is its antiderivative, then: \[ \int_{a}^{b} f(x) \, dx = F(b) - F(a). \]
  • The Second Part states that if \( f \) is continuous on \([a, b]\), then the function \( g(x) = \int_{a}^{x} f(t) \, dt \) is a continuous function on \([a, b]\) and differentiable on \((a, b)\), with \( g'(x) = f(x). \)

Although the theorem itself wasn't directly used for solving the given problem with equal limits, it's a foundation for understanding how integrals generally work. Mastery of the theorem allows leveraging integral properties for quick evaluations and solving more complex integrations easily in calculus.