Problem 51 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The integral evaluates to 0.

Step 1: Understand the Problem

The given integral is a definite integral, with both the upper and lower limits being the same, which are 2. The integral is \( \int_{2}^{2} \cos(3x^2) \, dx \).

Step 2: Apply Integral Rules

Recall a property of definite integrals: If the upper and lower limits of a definite integral are the same, then the value of the integral is zero. Therefore, \( \int_{a}^{a} f(x) \, dx = 0 \), where \( a \) is any constant.

Step 3: Conclusion

Applying the rule from Step 2 to \( \int_{2}^{2} \cos(3x^2) \, dx \), we conclude that the integral evaluates to 0 since both integration limits are equal.

Key Concepts

Integral Calculus

Integral calculus is a branch of mathematics focused on the accumulation of quantities, such as areas under a curve or total change over an interval. It involves concepts such as definite integrals, which have specific upper and lower limits and represent the net area between the curve and the x-axis from one point to another. This field of study is essential for solving problems related to areas, volumes, displacement, and other physical quantities.

• Definite Integrals: These integrals are used to compute the precise accumulation of a quantity between two boundaries.
• Indefinite Integrals: Unlike definite integrals, indefinite integrals do not have set limits and instead include a constant of integration to account for all potential accumulation paths.

Integral calculus is critical in understanding how to calculate diverse properties in physical and mathematical scenarios, using both definite and indefinite integrals.

Integral Properties

In the realm of integral calculus, integral properties equip us with certainties about integrals without the need to perform direct computation. These properties guide us in manipulating integrals easily.
Some fundamental properties include:

• Linearity: For any constants \(a\) and \(b\), we can state that \[ \int (a f(x) + b g(x)) \, dx = a \int f(x) \, dx + b \int g(x) \, dx \]
• Bounds Equality: This property appears in our exercise. If the limits of integration are the same, the integral is always zero: \( \int_{a}^{a} f(x) \, dx = 0 \).
• Additivity: If you break an integral into parts, you can add the segments: \( \int_{a}^{c} f(x) \, dx = \int_{a}^{b} f(x) \, dx + \int_{b}^{c} f(x) \, dx \).

These properties make it much easier to evaluate and manipulate integrals by bypassing complex computational steps.

Evaluation of Integrals

Evaluating integrals involves applying rules and techniques to find the value of an integral. For definite integrals, this often means calculating the exact area under the curve described by the function.
In our specific exercise example, evaluating the integral \( \int_{2}^{2} \cos(3x^2) \, dx \) is straightforward due to integral properties. Here, because the upper and lower bounds are identical (both 2), there's no area to calculate. As such, according to the property of bounds equality, the evaluated integral results in zero.
This example demonstrates how understanding properties can instantly lead to solutions without complex calculations. Recognizing such integral properties proves invaluable in simplifying the evaluation process in integral calculus.