Problem 13 - Applied Calculus

Question

(a) Use a graph of the integrand to make a rough estimate of the integral. Explain your reasoning. (b) Use a computer or calculator to find the value of the definite integral. $$ \int_{0}^{3} \sqrt{x} d x $$

Step-by-Step Solution

Answer

The integral \( \int_{0}^{3} \sqrt{x} \, dx \) evaluates to approximately 3.464.

Step 1: Understand the Integrand

The integrand of the given problem is \( f(x) = \sqrt{x} \). This is a non-linear function, specifically the square root function, which is increasing and concave down for \( x \geq 0 \).

Step 2: Sketch the Graph

To make a rough estimate of the integral, sketch the graph of \( y = \sqrt{x} \) from \( x = 0 \) to \( x = 3 \). Observing the graph, you will see it starts at the origin \((0,0)\) and rises gently, passing through \( (1,1) \) and \( (4,2) \), but we are interested in \( x = 0 \) to \( x = 3 \).

Step 3: Estimate the Area

The area under \( f(x) = \sqrt{x} \) from \( x = 0 \) to \( x = 3 \) can be estimated by approximating the area under the curve with geometric shapes. At \( x = 3 \), \( \sqrt{3} \approx 1.73 \), which provides an upper limit to the height. The area can be visualized as falling between a triangle with area \( \frac{1}{2} \times 3 \times 1.73 \approx 2.6 \) and a rectangle with area \( 3 \times 1 = 3 \). Thus, the rough estimate is between 2.6 and 3.

Step 4: Calculate the Exact Integral

Use a calculator to compute the integral \( \int_{0}^{3} \sqrt{x} \, dx \). The antiderivative of \( \sqrt{x} \) is \( \frac{2}{3}x^{3/2} \). Therefore, the integral evaluates to: \[ \int_{0}^{3} \sqrt{x} \, dx = \left[ \frac{2}{3}x^{3/2} \right]_{0}^{3} = \frac{2}{3}(3^{3/2}) - \frac{2}{3}(0^{3/2}) \]Calculating gives \[ \frac{2}{3} \times (3 \sqrt{3}) = 2 \times \sqrt{3} \approx 2 \times 1.732 = 3.464 \].

Step 5: Compare the Estimate and Exact Result

The rough estimate was between 2.6 and 3, based on a visual approximation, and the exact result is approximately 3.464. This shows that calculations done by the calculator are more precise, accounting for the true curve of \( \sqrt{x} \) as opposed to linear approximations.

Key Concepts

Square Root Function

The square root function, denoted as \( f(x) = \sqrt{x} \), is an important non-linear function in calculus. It is defined only for non-negative values of \( x \). As \( x \) increases, \( \sqrt{x} \) also increases, but at a decreasing rate. This means the function is described as concave down.

• At \( x = 0 \), the function starts at the origin point \( (0,0) \).
• It continues to rise gently, passing through points like \( (1,1) \) and reaching points like \( (3,1.73) \).
• Unlike linear functions, the vertical change (height) increases less than the horizontal change (base), making the curve appear as smooth and gradual.

Understanding the behavior of the square root function is crucial when evaluating definite integrals, as it influences the area under the curve estimation.

Antiderivative

The antiderivative is a fundamental concept in calculus, representing a function whose derivative is the given function. In other words, if you have a function like \( f(x) = \sqrt{x} \), the antiderivative is a function \( F(x) \) such that \( F'(x) = f(x) \).
For the square root function, the antiderivative is \( F(x) = \frac{2}{3}x^{3/2} \). Here's how this works:

• When you differentiate \( F(x) = \frac{2}{3}x^{3/2} \), you get back \( \sqrt{x} \), confirming that it's the correct antiderivative.
• This allows us to compute the definite integral over a specific interval.
• For instance, when evaluating \( \int_{0}^{3} \sqrt{x} \, dx \), we find the difference \( F(3) - F(0) \).

Mastering antiderivatives leads to accurate integration results, essential for precise calculus analyses.

Graph Estimation

Graph estimation is a useful technique for approximating the value of a definite integral when an exact calculation isn't feasible. By sketching or visualizing the graph of the function, one can make reasonable estimates about the area under the curve.
To estimate the integral of \( \sqrt{x} \) from 0 to 3:

• Plot the curve \( y = \sqrt{x} \) on a graph, noting how it behaves between \( x = 0 \) and \( x = 3 \).
• Identify simple geometric shapes such as triangles and rectangles to approximate the area under the curve.
• The area under the curve from 0 to 3 appears to be between a smaller triangle and a larger rectangle, giving an estimate range of 2.6 to 3.

Although this method lacks precision, it gives a quick and intuitive understanding of the integral's magnitude.

Calculus Techniques

Calculus offers a variety of techniques for evaluating integrals, both analytically and numerically. These methods ensure comprehensive understanding and accurate calculations:

• Finding the antiderivative is often the first approach, like using \( \frac{2}{3}x^{3/2} \) for \( \sqrt{x} \).
• We calculate definite integrals by using the Fundamental Theorem of Calculus, which applies the antiderivative over the limits: \( a \) to \( b \).
• Numerical solutions can be sought via calculators, ensuring precise values, as demonstrated by evaluating \( \int_{0}^{3} \sqrt{x} \, dx \) to get approximately 3.464.

These calculus tools are essential for solving real-world problems involving area, volume, and other quantities.