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:

1. ∫(-6)^7 x^2 dx = (x^3)/3 |(-6)^7 = (7^3 - (-6)^3)/3 = 343 - (-216)/3 = 186
2. ∫(-6)^7 2x dx = 2(x^2/2) |(-6)^7 = 2(7^2 - (-6)^2)/2 = 98 - 72 = 26
3. ∫(-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.