Question

Find the area between the curves y = x² and y = 3x² - 11x for 1 ≤ x ≤ 2.

a) 1/3 square units
b) 1 square unit
c) 2 square units
d) 4 square units

Answer 1

To find the area between the curves y = x² and y = 3x² - 11x, we need to calculate the definite integral of the difference between the two functions over the given interval. The area between the curves is 1 square unit.

Step-by-step explanation:

To find the area between the curves y = x² and y = 3x² - 11x for 1 ≤ x ≤ 2, we need to calculate the definite integral of the difference between the two functions over the given interval.

The area can be found by evaluating the integral:

A = ∫[1, 2] (3x² - 11x - x²) dx

Using the power rule for integration, we can simplify the integral and solve for the area:

A = ∫[1, 2] (2x² - 11x) dx

Calculating the definite integral, we get:

A = [2x³/3 - (11x²)/2] | [1, 2]

A = (16/3 - 22) - (2/3 - 11/2)

A = 1 square unit