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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.
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Practice Questions
1 question
Q1
Find 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 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.
Questions & Step-by-step Solutions
1 item
Q
Q: Find the area under the curve y = x^2 from x = 1 to x = 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.
Steps: 8
Show Steps
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.
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