Calculate the area between the curves y = x and y = x^2 from x = 0 to x = 1.

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Question: Calculate the area between the curves y = x and y = x^2 from x = 0 to x = 1.

Options:

Correct Answer: 0.5

Solution:

The area is given by the integral from 0 to 1 of (x - x^2) dx. This evaluates to [x^2/2 - x^3/3] from 0 to 1 = (1/2 - 1/3) = 1/6 = 0.5.

Calculate the area between the curves y = x and y = x^2 from x = 0 to x = 1.

Calculate the area between the curves y = x and y = x^2 from x = 0 to x = 1.

Correct Answer: 0.1667

Step 1: Identify the curves. We have two curves: y = x (a straight line) and y = x^2 (a parabola).
Step 2: Determine the points of intersection. Set x = x^2 to find where the curves meet. This gives us x(x - 1) = 0, so the points of intersection are at x = 0 and x = 1.
Step 3: Set up the integral. The area between the curves from x = 0 to x = 1 is found by integrating the difference of the two functions: (x - x^2).
Step 4: Write the integral. We need to calculate the integral from 0 to 1 of (x - x^2) dx.
Step 5: Calculate the integral. The integral of (x - x^2) is (x^2/2 - x^3/3).
Step 6: Evaluate the integral from 0 to 1. Substitute 1 into the expression: (1^2/2 - 1^3/3) = (1/2 - 1/3).
Step 7: Simplify the result. To subtract 1/3 from 1/2, find a common denominator (which is 6): (3/6 - 2/6) = 1/6.
Step 8: State the final area. The area between the curves from x = 0 to x = 1 is 1/6.

Area between curves – Calculating the area between two curves involves finding the integral of the difference between the two functions over a specified interval.
Definite integrals – Understanding how to evaluate definite integrals and apply the Fundamental Theorem of Calculus.
Function intersection – Identifying the points where the two curves intersect to determine the limits of integration.