Problem 1 - Applied Calculus

Question

Find the area under the graph of \(f(x)=x^{2}+2\) between \(x=0\) and \(x=6.\)

Step-by-Step Solution

Answer

The area under the curve is 84 square units.

Step 1: Set up the integral

To find the area under the curve of the function \(f(x) = x^2 + 2\) between \(x = 0\) and \(x = 6\), we need to evaluate the definite integral \(\int_{0}^{6} (x^2 + 2) \, dx\).

Step 2: Integrate the function

Perform the integration of the function \(x^2 + 2\). The integral of \(x^2\) is \(\frac{x^3}{3}\) and the integral of \(2\) is \(2x\). Therefore, the antiderivative of \(x^2 + 2\) is \(\frac{x^3}{3} + 2x\).

Step 3: Evaluate the definite integral

Substitute the limits of integration into the antiderivative. First, evaluate it at \(x = 6\): \(\frac{6^3}{3} + 2 \cdot 6 = 72 + 12 = 84\). Then evaluate it at \(x = 0\): \(\frac{0^3}{3} + 2 \cdot 0 = 0\).

Step 4: Compute the area

Subtract the result of the antiderivative evaluated at \(x = 0\) from the result evaluated at \(x = 6\): \(84 - 0 = 84\). Thus, the area under the curve from \(x = 0\) to \(x = 6\) is 84.

Key Concepts

Antiderivative

The concept of an antiderivative is a crucial part of calculus. An antiderivative, also known as an indefinite integral, is a function whose derivative is the original function. For example, when finding the antiderivative of a function like \(x^2 + 2\), we're looking for another function that, when differentiated, gives us \(x^2 + 2\).
In this particular example:

• The antiderivative of \(x^2\) is \(\frac{x^3}{3}\).
• The antiderivative of \(2\) is \(2x\).

Thus, the antiderivative (or the general form of the integral) for \(x^2 + 2\) is \(\frac{x^3}{3} + 2x + C\), where \(C\) is a constant. When dealing with definite integrals, the constant \(C\) cancels out, as the specific values of the function at the boundary points are what matters.

Area Under a Curve

Finding the area under a curve is a common application of integral calculus. The area under a function's curve between two points on the \(x\)-axis can be calculated using the definite integral. This method is particularly useful when the function is difficult to sum up geometrically.
For instance, to find the area under the curve of \(f(x) = x^2 + 2\) from \(x = 0\) to \(x = 6\), we calculate the definite integral \(\int_{0}^{6} (x^2 + 2) \, dx\). This gives a numerical value representing the space between the curve and the \(x\)-axis between those two points.
Hence, after computing this integral, we find that the area under the curve is 84 square units. The process involves calculating the antiderivative, substituting the limits of integration, and subtracting the results.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a key pillar in calculus, connecting the concept of differentiation with integration. It consists of two main parts that provide an incredibly powerful tool for evaluating definite integrals:

• **First part**: It states that if a function is continuous over an interval, then the integral of the function over that interval can be found using its antiderivative.
• **Second part**: It asserts that the derivative of the integral of a function is the original function itself.

In practical terms, these principles allow us to evaluate definite integrals by finding an antiderivative of the function, and then using this antiderivative to find the difference at the limits of integration.
For our example of \(f(x) = x^2 + 2\), the Fundamental Theorem of Calculus lets us take the antiderivative \(\frac{x^3}{3} + 2x\), evaluate it at the limits 6 and 0, and subtract the results. This yielding an area of 84, showcases its efficiency and importance in solving such integral problems.