Finding the Total Area Between the Curve y = x^2 + 2x - 3 and the X-Axis
As an excellent high school mathematics teacher, I'm excited to guide you through the process of finding the total area between the curve y = x^2 + 2x - 3 and the x-axis, for the interval x = -6 to x = 7. We'll also evaluate the integral ∫(-6)^7 f(x) dx and interpret the value in terms of the area.
Visualizing the Curve and the X-Axis
To begin, let's visualize the curve y = x^2 + 2x - 3 and the x-axis on a coordinate plane. The curve is a parabolic function that opens upwards, and the x-axis is the horizontal line at y = 0.
The region of interest is the area bounded by the curve, the x-axis, and the vertical lines at x = -6 and x = 7. This area represents the total area between the curve and the x-axis for the given interval.
Finding the Total Area
To find the total area between the curve and the x-axis, we can use the concept of integration. The area can be calculated by integrating the function f(x) = x^2 + 2x - 3 over the interval [-6, 7].
The formula for the area is:
Area = ∫(-6)^7 f(x) dx
Where f(x) = x^2 + 2x - 3.
To evaluate the integral, we can use the power rule of integration:
∫x^n dx = (x^(n+1))/(n+1) + C
Applying the power rule to our function, we get:
∫(-6)^7 (x^2 + 2x - 3) dx = ∫(-6)^7 x^2 dx + ∫(-6)^7 2x dx - ∫(-6)^7 3 dx
Evaluating each integral:
∫(-6)^7 x^2 dx = (x^3)/3 |(-6)^7 = (7^3 - (-6)^3)/3 = 343 - (-216)/3 = 186∫(-6)^7 2x dx = 2(x^2/2) |(-6)^7 = 2(7^2 - (-6)^2)/2 = 98 - 72 = 26∫(-6)^7 3 dx = 3(x) |(-6)^7 = 3(7 - (-6)) = 39
Substituting these values, we get:
Area = 186 + 26 - 39 = 173
Therefore, the total area between the curve y = x^2 + 2x - 3 and the x-axis, for the interval x = -6 to x = 7, is 173 square units.
Interpreting the Integral Value
The integral ∫(-6)^7 f(x) dx represents the signed area under the curve y = x^2 + 2x - 3 over the interval [-6, 7]. The value of the integral, 173, is the net area between the curve and the x-axis.
This value can be interpreted as follows:
- The positive area above the x-axis is the area where the curve is above the x-axis, which corresponds to the region where
y = x^2 + 2x - 3 > 0. - The negative area below the x-axis is the area where the curve is below the x-axis, which corresponds to the region where
y = x^2 + 2x - 3 < 0. - The net area, 173, is the difference between the positive area and the negative area.
In other words, the integral ∫(-6)^7 f(x) dx gives us the total area between the curve and the x-axis, taking into account both the positive and negative regions. This value can be used to understand the overall behavior of the function and its relationship with the x-axis.