Question

Find the area between the curves:y=x³-11 v+28 x y=-x³+11 x²-28 x

Answer 1

To find the area between the curves, find the points of intersection and use definite integration.

Step-by-step explanation:

To find the area between the curves, we need to find the points of intersection first. Equate the two functions: y = x³ - 11x + 28 and y = -x³ + 11x² - 28x to solve for x. Once you find the x-values, integrate the difference between the two functions from the smaller x-value to the larger x-value to find the area.

Let's solve for the points of intersection:

1. Set x³ - 11x + 28 = -x³ + 11x² - 28x.
2. Rearrange the equation to form a quadratic equation: 2x³ - 11x² + 57x - 28 = 0.
3. Use a graphing calculator or factor the quadratic equation to find the x-values.
4. Integrate the difference of the two functions from the smaller x-value to the larger x-value: ∫[(x³ - 11x + 28) - (-x³ + 11x² - 28x)] dx.
5. Calculate the definite integral.

This will give you the area between the curves.