Find the area under the curve y = x^2 from x = 0 to x = 2.

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Question: Find the area under the curve y = x^2 from x = 0 to x = 2.

Options:

Correct Answer: 8/3

Solution:

The area under the curve y = x^2 from 0 to 2 is given by the integral ∫(from 0 to 2) x^2 dx = [x^3/3] from 0 to 2 = (2^3/3) - (0^3/3) = 8/3.

Find the area under the curve y = x^2 from x = 0 to x = 2.

Find the area under the curve y = x^2 from x = 0 to x = 2.

Step 1: Identify the function you want to find the area under. In this case, the function is y = x^2.
Step 2: Determine the limits of integration. We want to find the area from x = 0 to x = 2.
Step 3: Set up the integral to calculate the area. This is written as ∫(from 0 to 2) x^2 dx.
Step 4: Find the antiderivative of x^2. The antiderivative is (x^3)/3.
Step 5: Evaluate the antiderivative at the upper limit (x = 2) and the lower limit (x = 0).
Step 6: Calculate the value at the upper limit: (2^3)/3 = 8/3.
Step 7: Calculate the value at the lower limit: (0^3)/3 = 0.
Step 8: Subtract the lower limit value from the upper limit value: (8/3) - (0) = 8/3.
Step 9: The area under the curve from x = 0 to x = 2 is 8/3.

Definite Integral – The process of calculating the area under a curve between two specified points using integration.
Polynomial Functions – Understanding the properties and behavior of polynomial functions, specifically quadratic functions in this case.
Fundamental Theorem of Calculus – Connecting differentiation and integration, allowing the evaluation of definite integrals using antiderivatives.