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Calculate ∫ from 0 to 1 of (4x^3 - 4x^2 + 1) dx.
Calculate ∫ from 0 to 1 of (4x^3 - 4x^2 + 1) dx.
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Practice Questions
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Q1
Calculate ∫ from 0 to 1 of (4x^3 - 4x^2 + 1) dx.
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The integral evaluates to [x^4 - (4/3)x^3 + x] from 0 to 1 = 1 - (4/3) + 1 = 2/3.
Questions & Step-by-step Solutions
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Q
Q: Calculate ∫ from 0 to 1 of (4x^3 - 4x^2 + 1) dx.
Solution:
The integral evaluates to [x^4 - (4/3)x^3 + x] from 0 to 1 = 1 - (4/3) + 1 = 2/3.
Steps: 7
Show Steps
Step 1: Identify the function to integrate, which is (4x^3 - 4x^2 + 1).
Step 2: Find the antiderivative of the function. The antiderivative is x^4 - (4/3)x^3 + x.
Step 3: Evaluate the antiderivative at the upper limit (1) and the lower limit (0).
Step 4: Calculate the value at the upper limit: (1^4 - (4/3)(1^3) + 1) = 1 - (4/3) + 1.
Step 5: Calculate the value at the lower limit: (0^4 - (4/3)(0^3) + 0) = 0.
Step 6: Subtract the lower limit value from the upper limit value: (1 - (4/3) + 1) - 0 = 2 - (4/3).
Step 7: Simplify the result: 2 - (4/3) = (6/3) - (4/3) = (2/3).
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