Problem 38 - Applied Calculus

Question

Find the area between the curve \(y=x^{2}+3\) and the line \(y=2 x\) (shown below) from \(x=0\) to \(x=3\).

Step-by-Step Solution

Answer

The area between the curves from \(x=0\) to \(x=3\) is 9 square units.

Step 1: Set Up the Problem

To find the area between the two curves, we first need to consider the functions given: the curve \(y = x^2 + 3\) and the line \(y = 2x\). We will integrate the difference of the functions over the interval \([0, 3]\).

Step 2: Determine the Intersection Points

To determine the limits for integration or verify any intersections, set \(x^2 + 3 = 2x\). Simplifying this gives \(x^2 - 2x + 3 = 0\). Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1, b = -2, c = 3\). However, solving it we notice there are no real intersections over the range \([0, 3]\) since the discriminant \(b^2 - 4ac = -8\) is negative.

Step 3: Calculate the Area

Integrate the difference between the larger function (the line) and the smaller function (the curve) from \(x = 0\) to \(x = 3\): \[\int_0^3 [(2x) - (x^2 + 3)] \, dx = \int_0^3 (2x - x^2 - 3) \, dx.\]

Step 4: Simplify and Integrate

Simplify the integrand: \(2x - x^2 - 3\). Now, find the integral: \[\int (2x - x^2 - 3) \, dx = \left[ x^2 - \frac{x^3}{3} - 3x \right]_0^3.\] Evaluate this from 0 to 3: \[\left[ 3^2 - \frac{3^3}{3} - 3(3) \right] - \left[ 0 \right].\]

Step 5: Solve the Definite Integral

Calculate the values: \[9 - 9 - 9 = -9.\] The negative result indicates the function \(x^2 + 3\) is actually above the line \(2x\) over the interval, thus, the area can be expressed as 9 square units by taking the absolute value.

Key Concepts

Definite Integral

The concept of a definite integral is fundamental when calculating the area between curves. In this context, it helps us find the total "net" area between two functions over a given interval. This involves integrating the difference between the functions — specifically, subtracting the lower function from the higher one within the interval of interest.
The definite integral is represented by the notation \int_\{a\}^\{b\} f(x) \, dx, where \(a\) and \(b\) are the endpoints of the interval over which you are integrating. The integrand, \(f(x)\), is the function you want to integrate, and \, dx indicates integration with respect to \(x\). In the exercise provided, the difference between the two functions is integrated over the interval \[\[\begin{equation} \int_0^3 \left( (2x) - (x^2 + 3) \right) \, dx.\end{equation}\] \]

• The limits of integration \(0\) and \(3\) specify where to start and end the calculation.
• The integrand \((2x - x^2 - 3)\) represents the difference between the line and the curve.

The calculation of the definite integral involves finding an antiderivative of the integrand and then calculating its value at the endpoints \(0\) and \(3\). The definite integral essentially "adds up" small slices of area between the curves from one endpoint to the other.

Intersection Points

Finding intersection points is crucial when analyzing the area between curves because intersections help define the boundaries of the region of interest. The points where two functions cross each other tell us where we need to pay special attention — possibly changing which function is on top.
In the given problem, we need to find if the curve \(y = x^2 + 3\) and the line \(y = 2x\) intersect within the interval \([0, 3]\). This involves solving the equation:

• Equals setting: \(x^2 + 3 = 2x\)
• Rearranging gives the quadratic equation \(x^2 - 2x + 3 = 0\)

For our problem, upon attempting to solve this quadratic equation, we notice the discriminant is negative. Specifically,\[b^2 - 4ac = (-2)^2 - 4(1)(3) = 4 - 12 = -8\]Since this value is negative, it infers that there are no real solutions, meaning the line and the parabola do not intersect at any real point within the specified range. Thus, the higher function throughout the interval is consistently the same, simplifying the evaluation of the definite integral.

Quadratic Formula

The quadratic formula is a valuable tool used to find the roots of quadratic equations like \(ax^2 + bx + c = 0\). When solving for intersection points between functions, a common approach involves equating and rearranging the equations until only one variable remains.
In our task, we equated \(x^2 + 3 = 2x\) to form the quadratic equation \(x^2 - 2x + 3 = 0\), where you identify:

• \(a = 1\)
• \(b = -2\)
• \(c = 3\)

The quadratic formula helps us solve this equation by providing the roots as follows:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]However, upon calculation, the discriminant \(b^2 - 4ac\) turned out negative, specifically \(-8\) in this situation. A negative discriminant indicates no real roots, meaning no real intersection points exist. This informs us that the functions don't cross within the interval considered, simplifying any further calculations of the area.