Question: Evaluate the integral ∫(1 to 2) (3x^2 - 4) dx. (2019)
Options:
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Correct Answer: 1
Exam Year: 2019
Solution:
∫(1 to 2) (3x^2 - 4) dx = [x^3 - 4x] from 1 to 2 = (8 - 8) - (1 - 4) = 3.
Evaluate the integral ∫(1 to 2) (3x^2 - 4) dx. (2019)
Practice Questions
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Evaluate the integral ∫(1 to 2) (3x^2 - 4) dx. (2019)
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Questions & Step-by-Step Solutions
Evaluate the integral ∫(1 to 2) (3x^2 - 4) dx. (2019)
Step 1: Identify the integral to evaluate: ∫(1 to 2) (3x^2 - 4) dx.
Step 2: Find the antiderivative of the function 3x^2 - 4. The antiderivative is x^3 - 4x.
Step 3: Evaluate the antiderivative at the upper limit (x = 2): (2^3 - 4*2) = (8 - 8) = 0.
Step 4: Evaluate the antiderivative at the lower limit (x = 1): (1^3 - 4*1) = (1 - 4) = -3.
Step 5: Subtract the lower limit result from the upper limit result: 0 - (-3) = 0 + 3 = 3.
Step 6: Conclude that the value of the integral ∫(1 to 2) (3x^2 - 4) dx is 3.
Definite Integral Evaluation – The process of calculating the area under a curve defined by a function over a specific interval.
Fundamental Theorem of Calculus – This theorem connects differentiation and integration, allowing the evaluation of definite integrals using antiderivatives.
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