Problem 29 - Calculus and its applications

Question

Find the area of the region bounded by the graphs of the given equations. $$ y=x^{2}+3, y=x^{2}, x=1, x=2 $$

Step-by-Step Solution

Answer

The area of the region is 3 square units.

Step 1: Determine the Area Between the Curves

To find the area of the region bounded by the curves, consider the functions provided. The area we want to calculate is between the functions \( y = x^2 + 3 \) (upper curve) and \( y = x^2 \) (lower curve), bounded from \( x = 1 \) to \( x = 2 \).

Step 2: Set Up the Integral

The area \( A \) between the curves from \( x = 1 \) to \( x = 2 \) can be found by integrating the difference between the upper function and the lower function: \[A = \int_{1}^{2} ((x^2 + 3) - x^2) \, dx.\] Simplify the integrand: \[A = \int_{1}^{2} 3 \, dx.\]

Step 3: Integrate and Evaluate

Calculate the integral: \[A = \int_{1}^{2} 3 \, dx = 3x \Big|_{1}^{2}.\] Evaluate at the bounds: \[A = 3(2) - 3(1) = 6 - 3 = 3.\]

Step 4: Conclude the Area

The area of the region is calculated to be \(3\) square units.

Key Concepts

Area between curves

Finding the area between two curves is a fundamental task in calculus. When you have two functions and you're asked to find the area between them, you're essentially looking at the space enclosed between their graphs over a specific interval on the x-axis.
To approach this, visualize the two curves plotted on a coordinate plane. They will overlap in certain regions. The area between them can be thought of as the surface you would "paint" if you filled in the space between them within a given interval.
For curves given as functions of x, say \(f(x)\) as the upper curve and \(g(x)\) as the lower curve:

• Identify the region of interest by determining the interval \([a, b]\) on the x-axis where you want to measure the area.
• Set up an integral from \(a\) to \(b\) of the upper function minus the lower function: \[ \int_{a}^{b} (f(x) - g(x)) \, dx \]

This process simplifies to subtracting one function from the other and integrating over the set bounds.

Definite integral

A definite integral is a way to compute the accumulation of a quantity, such as area. It has both a start point and an end point along the x-axis, denoted by the limits of integration.
To find a definite integral:

• Identify the function to be integrated, called the integrand.
• Determine the limits of integration, which are the bounds \(a\) and \(b\).
• Compute the integral of the function with respect to x over this interval: \[ \int_{a}^{b} f(x) \, dx \]
• Evaluate the result by calculating the difference between the values of the antiderivative of the integrand at \(b\) and \(a\).

The definite integral accounts for both the shape and relative position of a function over an interval, often representing concepts such as total accumulated area or distance.

Fundamental theorem of calculus

The Fundamental Theorem of Calculus is a key principle linking the concepts of differentiation and integration. It asserts that these two operations are essentially inverses of each other.
This theorem is usually divided into two parts:

• The first part allows you to find the derivative of an integral function, emphasizing that the process of differentiation undoes integration.
• The second part states that if you have an antiderivative \(F(x)\) of a continuous function \(f(x)\), then the definite integral of \(f(x)\) from \(a\) to \(b\) is given by: \[ F(b) - F(a) \]

This dual role of the Fundamental Theorem simplifies solving problems involving area and accumulation, providing a procedural method to evaluate definite integrals efficiently.