Final answer:
To find he area bounded between the curve f(x)=eˣ −3 and the line g(x)=4x−2x over the interval [−1,4] is 13/2 square units.
The answer is option ⇒b
Step-by-step explanation:
To find the area bounded between the curve f(x) = eˣ - 3 and the line g(x) = 4x - 2x over the interval [-1, 4], we need to calculate the definite integral of the absolute difference between the two functions over that interval.
1. Find the absolute difference between the two functions: |f(x) - g(x)|.
- The absolute difference between the two functions is |(eˣ - 3) - (4x - 2x)| = |eˣ - 3 - 2x|.
2. Set up the definite integral:
- The definite integral of |eˣ - 3 - 2x| over the interval [-1, 4] is given by:
∫[from -1 to 4] |eˣ - 3 - 2x| dx.
3. Evaluate the definite integral:
- The integral of |eˣ - 3 - 2x| can be split into two parts:
∫[from -1 to 0] (3 - 2x - eˣ) dx + ∫[from 0 to 4] (eˣ - 3 - 2x) dx.
- Evaluating the definite integrals using the antiderivative, we get:
(3x - x² - eˣ) evaluated from -1 to 0 + (eˣ - 3x - x²/2) evaluated from 0 to 4.
- Simplifying further, we get:
(3(0) - (0)² - e⁰) - (3(-1) - (-1)² - e⁻¹) + (e⁴ - 3(4) - (4)²/2) - (e⁰ - 3(0) - (0)²/2).
- Simplifying the above expression, we get:
-1 + e⁻¹ + e⁴ - 13/2.
4. Calculate the final result:
- The final result of the definite integral is -1 + e⁻¹ + e⁴ - 13/2.
Therefore, the correct option is b) 13/2.
Hence, the area bounded between the curve f(x) = eˣ - 3 and the line g(x) = 4x - 2x over the interval [-1, 4] is 13/2.
The answer is option ⇒b.13/2