Math Problem Statement
jawab
∫[−1 to 2] (4x^3 + 2x^2 − 4) dx = 11. True or False?
Solution
The given integral problem is:
We need to evaluate this definite integral and verify if it equals or not.
Step 1: Integrate the function
The integral of with respect to is computed term by term: So, the antiderivative is:
Step 2: Apply the limits of integration
Using the Fundamental Theorem of Calculus, we evaluate at the limits and :
Evaluate :
= 16 + \frac{16}{3} - 8 = \frac{48}{3} + \frac{16}{3} - \frac{24}{3} = \frac{40}{3}.$$ #### Evaluate $$F(-1)$$: $$F(-1) = (-1)^4 + \frac{2(-1)^3}{3} - 4(-1) = 1 + \frac{-2}{3} + 4 = \frac{3}{3} - \frac{2}{3} + \frac{12}{3} = \frac{13}{3}.$$ #### Compute $$F(2) - F(-1)$$: $$F(2) - F(-1) = \frac{40}{3} - \frac{13}{3} = \frac{27}{3} = 9.$$ ### Step 3: Compare the result The computed value of the integral is $$9$$, not $$11$$. Therefore, the statement is **False**. --- Would you like further clarification or a step-by-step breakdown? Here are some related questions to explore: 1. How can definite integrals be used in real-world applications? 2. What is the Fundamental Theorem of Calculus, and how does it apply here? 3. How do you verify your result in definite integrals for correctness? 4. What techniques are useful for integrating polynomials? 5. Can you explain the importance of simplifying terms before evaluating limits? **Tip**: Always double-check your limits and computations when solving definite integrals!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Definite Integration
Polynomial Integration
Fundamental Theorem of Calculus
Formulas
∫ x^n dx = (x^(n+1))/(n+1) + C for n ≠ −1
Definite integral: ∫[a to b] f(x) dx = F(b) − F(a)
Theorems
Fundamental Theorem of Calculus
Suitable Grade Level
Grades 11-12