Area Between Curves: Step-by-Step Calculation

Hey guys! Today, we're diving deep into a classic calculus problem: finding the area bounded by two curves. Specifically, we'll be tackling the curves y = x² + 5x - 5 and y = x - 2. This might sound intimidating, but trust me, we'll break it down step-by-step, and by the end, you'll be a pro at solving these kinds of problems. We'll not only calculate the area but also learn how to visualize these curves by plotting their graphs. This will give us a clear picture of the region we are trying to find the area for.

Understanding the Problem: Area Between Curves

Before we jump into the math, let's make sure we understand the core concept. The area between curves represents the region enclosed by two or more functions graphed on a coordinate plane. Think of it as the space trapped between these curves. To find this area, we'll use the power of integral calculus. The basic idea is to integrate the difference between the two functions over the interval where they intersect. The function that is “on top” is subtracted from the function that is “on the bottom”. This ensures that the area we calculate is positive, as area is a scalar quantity.

The integral represents the accumulation of infinitesimally small areas, which, when summed up over the specified interval, gives us the total area between the curves. This method works because the integral essentially calculates the area under each curve separately, and then the subtraction gives us the area between them. It's like finding the area of a shape by breaking it down into infinitely thin rectangles and adding up their areas. This concept is fundamental in calculus and has wide applications in physics, engineering, and economics.

To successfully find the area between curves, it's crucial to first identify the points of intersection. These points define the limits of integration, which are the boundaries of the region we're interested in. If we don't find these intersection points accurately, our integration will be over the wrong interval, and the final area calculation will be incorrect. The intersection points are found by setting the two functions equal to each other and solving for x. These x-values will then be used as the limits of integration in our definite integral. So, accurately determining these points is the first and one of the most critical steps in the process. This is why we will start by finding these intersection points for the given curves.

Step 1: Finding the Points of Intersection

The first key step in solving this problem is to find where the two curves intersect. This is where the magic happens, as these points will define the boundaries of our integration. To find these points, we need to set the equations equal to each other:

y = x² + 5x - 5 y = x - 2

Setting them equal:

x² + 5x - 5 = x - 2

Now, let's rearrange the equation to get a quadratic equation in the standard form:

x² + 5x - x - 5 + 2 = 0 x² + 4x - 3 = 0

We now have a quadratic equation: x² + 4x - 3 = 0. To solve this, we can use the quadratic formula:

x = [-b ± √(b² - 4ac)] / 2a

Where a = 1, b = 4, and c = -3. Plugging these values into the formula, we get:

x = [-4 ± √(4² - 4 * 1 * -3)] / 2 * 1 x = [-4 ± √(16 + 12)] / 2 x = [-4 ± √28] / 2 x = [-4 ± 2√7] / 2 x = -2 ± √7

So, we have two x-values for the points of intersection:

x₁ = -2 + √7 x₂ = -2 - √7

These x-values are the x-coordinates of the points where the two curves intersect. Now we need to find the corresponding y-values. We can plug these x-values into either of the original equations. The simpler equation, y = x - 2, will be easier to work with. For each x-value, we'll calculate the corresponding y-value:

For x₁ = -2 + √7: y₁ = (-2 + √7) - 2 y₁ = -4 + √7

For x₂ = -2 - √7: y₂ = (-2 - √7) - 2 y₂ = -4 - √7

Thus, the points of intersection are (-2 + √7, -4 + √7) and (-2 - √7, -4 - √7). These points are crucial because they define the limits of our integration. Understanding these points geometrically is also important. They are the exact coordinates where the parabola and the line intersect, marking the beginning and the end of the region whose area we want to calculate. In the next step, we will use these points to set up our integral.

Step 2: Setting up the Integral

Now that we have the points of intersection, we can set up the integral to find the area. Remember, the area between two curves is given by the integral of the absolute difference between the functions, evaluated between the points of intersection. We first need to determine which function is