Question

In Exercises 35 and \(36,\) find the area of the region by subtracting the area of a triangular region from the area of a larger region. The region on or above the $$x$$ -axis bounded by the curves $$y=4-x^{2}$$ and $y=3 x$$

Short answer

After following the steps to find and evaluate the integrals, subtracting the smaller area from the larger one will give the area of the region bounded by the two curves, which must be the final answer.

Step-by-step solution

Find Intersection Points

To find the intersection points, we set the two functions equal to each other, i.e. \(4 - x^{2} = 3x\). Solving this quadratic equation will give the x-coordinates of the intersection points.

Solve the Quadratic Equation

The quadratic equation to solve is \(x^{2} + 3x - 4 = 0\). This can be factored into \((x - 1)(x + 4) = 0\), from which we get two solutions \(x = 1\) and \(x = -4\).

Setup the Integrals

The area under a curve from a to b is given by the integral \(\int_{a}^{b} f(x) dx\). The first integral is the area under \(4-x^2\) from -4 to 1 given by \(\int_{-4}^{1} (4-x^2) dx\), and the second integral is the area under \(3x\) from -4 to 1 given by \(\int_{-4}^{1} (3x) dx\).

Evaluate the Integrals

Evaluate the two integrals and subtract to find the area of the region. The indefinite integrals are given by \(-x^3/3 +4x\) for \(4-x^2\), and \(3x^2/2\) for \(3x\). Evaluate at the upper and lower bounds to find the definite integral, then subtract to find the area.

Calculate the Area

Subtract the smaller area under the straight line from the larger area under the curve. After doing the necessary computation, this will yield the final answer.

Key concepts

Quadratic Equations

Understanding quadratic equations is essential when finding the area between curves. A quadratic equation is of the form \(ax^2 + bx + c = 0\) and represents a parabola when plotted on a graph. To find the intersection points between two curves, we often need to solve a quadratic equation that arises from setting the equations of the two curves equal to each other.

In our example, the equation \(4 - x^2 = 3x\) turns into \(x^2 + 3x - 4 = 0\) after rearranging the terms. This specific equation can be factored easily, which provides a straightforward solution. However, not all quadratic equations can be factored simply, and other methods such as completing the square or using the quadratic formula may be necessary. Recognizing and solving quadratic equations are fundamental skills in algebra that are applied in calculus when finding areas between curves.

Integration

Integration is closely related to the concept of finding the area under a curve on a graph. It provides a way to calculate such areas by considering the curve as an infinite sum of infinitesimally thin rectangles and adding them together. The process of integration takes a function and returns its integral, which represents the area under the curve of the function between two points.

When finding the area between curves, we integrate the function that describes each curve. In our exercise, these functions are \(4 - x^2\) and \(3x\), representing a parabola and a straight line, respectively. Every integral represents the area under its respective curve. By integrating these functions over the interval marked by their intersection points, we can find the total area enclosed by the curves.

Definite Integrals

A definite integral is an integral with upper and lower bounds, which provides a numerical value representing the area under the curve of a function between those two points. To calculate a definite integral, we must find the antiderivative (or the indefinite integral) of the function first and then apply the Fundamental Theorem of Calculus by evaluating the antiderivative at the upper and lower limits and subtracting these values.

In practice, this means that for the area under \(4-x^2\) from \(x = -4\) to \(x = 1\), we find the indefinite integral and evaluate it at these bounds. Similarly, for the curve \(3x\), we find its indefinite integral and evaluate from \(x = -4\) to \(x = 1\) as well. The definite integrals are given by the difference between these evaluations, and their subtraction gives us the net area between the curves. This step is crucial in precisely calculating the area between curves.

Intersection Points

Intersection points are where two curves meet or cross each other on a graph. Finding intersection points is a critical first step when calculating the area between curves because these points define the limits of integration. The area between curves is essentially 'trapped' within the bounds established by the intersection points.

In our exercise, to find the intersection points of \(y = 4-x^2\) and \(y = 3x\) we set the equations equal and solve for \(x\) to determine where the curves intersect. In this case, solving \(4 - x^2 = 3x\) results in two intersection points at \(x = -4\) and \(x = 1\) that serve as the limits of integration. These points are critical for setting up the correct integrals to find the area between the specified curves.