Find the area under the curve y = x^2 from x = 0 to x = 3.

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Question: Find the area under the curve y = x^2 from x = 0 to x = 3.

Options:

Correct Answer: 9

Solution:

Area = ∫ from 0 to 3 of x^2 dx = [1/3 * x^3] from 0 to 3 = 9.

Find the area under the curve y = x^2 from x = 0 to x = 3.

Find the area under the curve y = x^2 from x = 0 to x = 3.

Correct Answer: 9

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 = 3.
Step 3: Set up the integral to calculate the area. This is written as ∫ from 0 to 3 of x^2 dx.
Step 4: Find the antiderivative of x^2. The antiderivative is (1/3) * x^3.
Step 5: Evaluate the antiderivative at the upper limit (x = 3) and the lower limit (x = 0).
Step 6: Calculate the value at the upper limit: (1/3) * (3^3) = (1/3) * 27 = 9.
Step 7: Calculate the value at the lower limit: (1/3) * (0^3) = (1/3) * 0 = 0.
Step 8: Subtract the lower limit value from the upper limit value: 9 - 0 = 9.
Step 9: The area under the curve from x = 0 to x = 3 is 9.

Definite Integral – The question tests the understanding of calculating the area under a curve using definite integrals.
Integration of Polynomial Functions – It assesses the ability to integrate polynomial functions, specifically x^2 in this case.
Fundamental Theorem of Calculus – The question requires applying the Fundamental Theorem of Calculus to evaluate the integral.