Question

Find the area of the region described below. The region bounded by y = 3x2 and y = 2x2 +1 The area of the region is (Type an integer or a simplified fraction.) Find the area of the region described. х The region bounded by y = |x - 4 and y = 3 х == is The area of the region bounded by y = |x-4 and y=- (Type an exact answer, using radicals as needed.)

Answer 1

To find the area of the region bounded by the curves y = 3x^2 and y = 2x^2 + 1, we integrate the difference between the two equations with respect to x and evaluate the integral between the points of intersection (-1 and 1). The area is 2/3.

Step-by-step explanation:

To find the area of the region bounded by the curves y = 3x^2 and y = 2x^2 + 1, we need to find the points of intersection first. Set the two equations equal to each other:

3x^2 = 2x^2 + 1

x^2 = 1

x = ±1

Now, we integrate the difference between the two equations with respect to x from -1 to 1:

Area = ∫((2x^2 + 1) - (3x^2)) dx

Area = ∫(-x^2 + 1) dx

Area = (-1/3)x^3 + x

Plugging in the bounds, we get:

Area = (-1/3)(1)^3 + (1) - [(-1/3)(-1)^3 + (-1)]

Area = -1/3 + 1 + 1/3 - 1

Area = 2/3