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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?
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Practice Questions
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Q1
What is the area under the curve y = x^2 from x = 1 to x = 3?
8/3
10/3
9/3
7/3
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The area is ∫(1 to 3) x^2 dx = [1/3 * x^3] from 1 to 3 = (27/3 - 1/3) = 26/3.
Questions & Step-by-step Solutions
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Q
Q: What is the area under the curve y = x^2 from x = 1 to x = 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.
Steps: 10
Show Steps
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.
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