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

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

Options:

Correct Answer: 10/3

Solution:

The area is given by the integral ∫ (x^2) dx from 1 to 3. This evaluates to [x^3/3] from 1 to 3 = (27/3 - 1/3) = 26/3.

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

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

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 = 1 to x = 3.
Step 3: Set up the integral to find the area. This is written as ∫ (x^2) dx from 1 to 3.
Step 4: Calculate the integral of x^2. The integral of x^2 is (x^3)/3.
Step 5: Evaluate the integral at the upper limit (x = 3). This gives (3^3)/3 = 27/3 = 9.
Step 6: Evaluate the integral at the lower limit (x = 1). This gives (1^3)/3 = 1/3.
Step 7: Subtract the lower limit result from the upper limit result. So, 9 - (1/3) = 9 - 0.3333 = 8.6667.
Step 8: Convert 8.6667 to a fraction. This is 26/3.

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.