91Ó°ÊÓ

Finding the intersection points of curves is a crucial step when determining the area between them. When two curves intersect, they share common points along the x-axis where their equations are equal. In other words, at these points, the y-values of both curves are the same.

To find these intersection points, we set the equations of the curves equal to each other. For the given exercise, we had two equations: \(y = 7 - 2x^2\) and \(y = x^2 + 4\). By solving \(7 - 2x^2 = x^2 + 4\), we found the solution to be \(3x^2 = 3\), which simplifies to \(x^2 = 1\).

This tells us that the curves intersect at \(x = -1\) and \(x = 1\).

• Consider solving each intersection point separately if you ever encounter a more complex equation.
• Check your work by plugging the x-values back into the original equations to ensure they yield the same y-value.

By knowing the intersection points, you can accurately define the limits for integration, which helps in finding the area between the curves.

Definite integrals are a foundational concept in calculus for determining the area under curves or between curves. In essence, a definite integral sums up an infinite number of infinitesimally small rectangles under a curve, giving you the total area.

When using definite integrals to find the area between curves, it's important first to decide which curve is on top and which is on the bottom across the interval in question. This decision dictates how you set up the integral: the top function minus the bottom function.

In our exercise, over the interval from \(-1\) to \(1\), the curve \(y = 7 - 2x^2\) is above \(y = x^2 + 4\). Therefore, the integral that represents the area between them is:

\[ \int_{-1}^{1} [(7 - 2x^2) - (x^2 + 4)] \, dx \]

• The upper and lower limits of the integral correspond to the intersection points you determined earlier.
• The integrand, which is the expression inside the integral, is simplified to make the calculation straightforward.

Remember, definite integrals provide the exact numerical value, representing the area of interest, when calculated correctly.

The area between two curves is a frequent application of integration in calculus. It involves calculating the region that lies between the graphs of two functions across a given interval.

To find this area, follow these steps:
1. Determine the intersection points for the limits of integration.
2. Identify which of the curves lies above the other within the interval.
3. Set up the definite integral of the upper curve minus the lower curve over that interval.

In the problem at hand, the function \(7 - 2x^2\) is above the function \(x^2 + 4\) from \(x = -1\) to \(x = 1\).

After setting up the integral, \(\int_{-1}^{1} (3 - 3x^2) \, dx\), calculate it to find

\[ Area = \left[ 3x - x^3 \right]_{-1}^{1} \]

Finally, substitute the points into the antiderivative and find the difference:

• Upper limit: \(3(1) - 1^3 = 2\)
• Lower limit: \(3(-1) - (-1)^3 = -4\)
• The area, therefore, is \(2 - (-4) = 4\).

Finding the area between curves involves these precise calculations and delivers the real-world application of calculus principles.