What is the area under the curve y = x^2 from x = 1 to x = 3?

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Question: What is the area under the curve y = x^2 from x = 1 to x = 3?

Options:

Correct Answer: 10/3

Solution:

The area is ∫(1 to 3) x^2 dx = [1/3 * x^3] from 1 to 3 = (27/3 - 1/3) = 26/3.

What is the area under the curve y = x^2 from x = 1 to x = 3?

What is the area under the curve y = x^2 from x = 1 to x = 3?

Step 1: Identify the function we want to find the area under, which is y = x^2.
Step 2: Determine the limits of integration, which are from x = 1 to x = 3.
Step 3: Set up the integral to find the area: ∫(1 to 3) x^2 dx.
Step 4: Find the antiderivative of x^2, which is (1/3) * x^3.
Step 5: Evaluate the antiderivative at the upper limit (x = 3): (1/3) * (3^3) = (1/3) * 27 = 9.
Step 6: Evaluate the antiderivative at the lower limit (x = 1): (1/3) * (1^3) = (1/3) * 1 = 1/3.
Step 7: Subtract the lower limit result from the upper limit result: 9 - (1/3).
Step 8: Convert 9 to a fraction with a denominator of 3: 9 = 27/3.
Step 9: Perform the subtraction: (27/3) - (1/3) = (27 - 1)/3 = 26/3.
Step 10: The area under the curve from x = 1 to x = 3 is 26/3.

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.