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What is the area under the curve y = x^2 from x = 0 to x = 3?

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Question: What is the area under the curve y = x^2 from x = 0 to x = 3?

Options:

  1. 9
  2. 6
  3. 4.5
  4. 3

Correct Answer: 9

Solution: 

The area is given by the integral ∫_0^3 x^2 dx = [x^3/3]_0^3 = 27/3 = 9.

What is the area under the curve y = x^2 from x = 0 to x = 3?

Practice Questions

Q1
What is the area under the curve y = x^2 from x = 0 to x = 3?
  1. 9
  2. 6
  3. 4.5
  4. 3

Questions & Step-by-Step Solutions

What is the area under the curve y = x^2 from x = 0 to x = 3?
Correct Answer: 9
  • Step 1: Understand that we want to find the area under the curve of the function y = x^2 from x = 0 to x = 3.
  • Step 2: We use the concept of integration to find this area. The integral we need to calculate is ∫_0^3 x^2 dx.
  • Step 3: To solve the integral, we find the antiderivative of x^2. The antiderivative is (x^3)/3.
  • Step 4: We will evaluate this antiderivative from the lower limit (0) to the upper limit (3). This is written as [x^3/3] from 0 to 3.
  • Step 5: First, we calculate the value at the upper limit (x = 3): (3^3)/3 = 27/3 = 9.
  • Step 6: Next, we calculate the value at the lower limit (x = 0): (0^3)/3 = 0/3 = 0.
  • Step 7: Now, we subtract the lower limit value from the upper limit value: 9 - 0 = 9.
  • Step 8: Therefore, the area under the curve y = x^2 from x = 0 to x = 3 is 9.
  • Definite Integral – The area under a curve between two points is calculated using definite integrals.
  • Polynomial Functions – Understanding the properties and behavior of polynomial functions, specifically quadratic functions like y = x^2.
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