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

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

Options:

Correct Answer: 6

Solution:

The area under the curve is given by ∫(from 0 to 2) (x^2 + 2x) dx = [x^3/3 + x^2] from 0 to 2 = (8/3 + 4) = 20/3.

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

Calculate the area under the curve y = x^2 + 2x 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 + 2x.
Step 2: Set up the integral to calculate the area under the curve from x = 0 to x = 2. This is written as ∫(from 0 to 2) (x^2 + 2x) dx.
Step 3: Find the antiderivative (the integral) of the function x^2 + 2x. The antiderivative is (x^3/3 + x^2).
Step 4: Evaluate the antiderivative at the upper limit (x = 2) and the lower limit (x = 0).
Step 5: Calculate the value at the upper limit: (2^3/3 + 2^2) = (8/3 + 4).
Step 6: Calculate the value at the lower limit: (0^3/3 + 0^2) = 0.
Step 7: Subtract the lower limit value from the upper limit value: (8/3 + 4) - 0 = 8/3 + 4.
Step 8: Convert 4 into a fraction with a denominator of 3: 4 = 12/3.
Step 9: Add the two fractions: (8/3 + 12/3) = 20/3.
Step 10: The area under the curve from x = 0 to x = 2 is 20/3.

Definite Integral – The process of calculating the area under a curve between two specified points using integration.
Polynomial Functions – Understanding how to integrate polynomial functions, specifically quadratic functions in this case.
Fundamental Theorem of Calculus – Applying the theorem to evaluate the definite integral by finding the antiderivative.