Find the area under the curve y = 3x^2 from x = 1 to x = 2.

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Question: Find the area under the curve y = 3x^2 from x = 1 to x = 2.

Options:

Correct Answer: 6

Solution:

The area under the curve is given by ∫(from 1 to 2) 3x^2 dx = [x^3] from 1 to 2 = (8 - 1) = 7.

Find the area under the curve y = 3x^2 from x = 1 to x = 2.

Find the area under the curve y = 3x^2 from x = 1 to x = 2.

Step 1: Identify the function we want to find the area under, which is y = 3x^2.
Step 2: Set up the integral to find the area from x = 1 to x = 2. This is written as ∫(from 1 to 2) 3x^2 dx.
Step 3: Calculate the integral of 3x^2. The integral of 3x^2 is x^3 (using the power rule of integration).
Step 4: Evaluate the integral from the lower limit (1) to the upper limit (2). This means we will calculate [x^3] from 1 to 2.
Step 5: Substitute the upper limit (2) into x^3: (2^3) = 8.
Step 6: Substitute the lower limit (1) into x^3: (1^3) = 1.
Step 7: Subtract the value at the lower limit from the value at the upper limit: 8 - 1 = 7.
Step 8: The area under the curve from x = 1 to x = 2 is 7.

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.