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

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

Options:

  1. 3.5
  2. 4
  3. 5
  4. 6

Correct Answer: 4

Solution: 

The area under the curve y = x^3 from x = 1 to x = 2 is given by ∫(from 1 to 2) x^3 dx = [x^4/4] from 1 to 2 = (16/4) - (1/4) = 4 - 0.25 = 3.75.

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

Practice Questions

Q1
What is the area under the curve y = x^3 from x = 1 to x = 2?
  1. 3.5
  2. 4
  3. 5
  4. 6

Questions & Step-by-Step Solutions

What is the area under the curve y = x^3 from x = 1 to x = 2?
Correct Answer: 3.75
  • Step 1: Identify the function we want to find the area under, which is y = x^3.
  • Step 2: Determine the limits of integration, which are from x = 1 to x = 2.
  • Step 3: Set up the integral to find the area: ∫(from 1 to 2) x^3 dx.
  • Step 4: Calculate the antiderivative of x^3, which is (x^4)/4.
  • Step 5: Evaluate the antiderivative at the upper limit (x = 2): (2^4)/4 = 16/4 = 4.
  • Step 6: Evaluate the antiderivative at the lower limit (x = 1): (1^4)/4 = 1/4 = 0.25.
  • Step 7: Subtract the lower limit result from the upper limit result: 4 - 0.25.
  • Step 8: Calculate the final result: 4 - 0.25 = 3.75.
  • Definite Integral – The question tests the understanding of calculating the area under a curve using definite integrals.
  • Polynomial Functions – It involves working with polynomial functions, specifically cubic functions in this case.
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