Determine the area between the curves y = x^3 and y = x from x = 0 to x = 1.

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Question: Determine the area between the curves y = x^3 and y = x from x = 0 to x = 1.

Options:

Correct Answer: 1/4

Solution:

The area is given by the integral from 0 to 1 of (x - x^3) dx. This evaluates to [x^2/2 - x^4/4] from 0 to 1 = (1/2 - 1/4) = 1/4.

Determine the area between the curves y = x^3 and y = x from x = 0 to x = 1.

Determine the area between the curves y = x^3 and y = x from x = 0 to x = 1.

Step 1: Identify the curves. We have two curves: y = x^3 and y = x.
Step 2: Find the points where the curves intersect. Set x^3 = x and solve for x. This gives us x(x^2 - 1) = 0, so x = 0 and x = 1.
Step 3: Determine which curve is on top between x = 0 and x = 1. For x in (0, 1), y = x is above y = x^3.
Step 4: Set up the integral to find the area between the curves. The area A is given by A = ∫ from 0 to 1 of (top curve - bottom curve) dx, which is A = ∫ from 0 to 1 of (x - x^3) dx.
Step 5: Calculate the integral. First, find the antiderivative of (x - x^3), which is (x^2/2 - x^4/4).
Step 6: Evaluate the antiderivative from 0 to 1. Plug in 1: (1^2/2 - 1^4/4) = (1/2 - 1/4).
Step 7: Simplify the result. (1/2 - 1/4) = 1/4.
Step 8: State the final answer. The area between the curves from x = 0 to x = 1 is 1/4.

Area Between Curves – The concept involves finding the area enclosed between two functions by integrating the difference of the functions over a specified interval.
Definite Integrals – Understanding how to evaluate definite integrals and apply the Fundamental Theorem of Calculus to find the area.
Function Intersection – Identifying the points where the two curves intersect, which is crucial for determining the limits of integration.