The area of the region bounded by `x=1, x=2, y=4x+1, y=e^(x)` is

Options

A. `7+e-e^(2)`
B. `5+e-e^(2)`
C. `7-e+e^(2)`
D. `7-2e+e^(2)`

Answer

To find the area of the region bounded by the curves \(x=1\), \(x=2\), \(y=4x+1\), and \(y=e^x\), we will follow these steps:

### Step 1: Identify the curves and their intersection points
We have two curves:
1. \(y = 4x + 1\)
2. \(y = e^x\)

We need to find the points of intersection of these two curves to understand the area we are dealing with.

### Step 2: Set the equations equal to each other
To find the intersection points, we set \(4x + 1 = e^x\).

### Step 3: Solve for \(x\)
This equation does not have a simple algebraic solution, so we can find the intersection points graphically or numerically. However, we can evaluate the functions at \(x=1\) and \(x=2\):
- At \(x=1\):
- \(y = 4(1) + 1 = 5\)
- \(y = e^1 \approx 2.718\)
- At \(x=2\):
- \(y = 4(2) + 1 = 9\)
- \(y = e^2 \approx 7.389\)

From this, we can see that the line \(y = 4x + 1\) is above the curve \(y = e^x\) between \(x=1\) and \(x=2\).

### Step 4: Set up the integral for the area
The area \(A\) between the curves from \(x=1\) to \(x=2\) can be calculated using the formula:
\[ A = \int_{1}^{2} [(4x + 1) - e^x] \, dx \]

### Step 5: Compute the integral
Now we will compute the integral:
\[ A = \int_{1}^{2} (4x + 1) \, dx - \int_{1}^{2} e^x \, dx \]

Calculating the first integral:
\[ \int (4x + 1) \, dx = 2x^2 + x \]
Evaluating from 1 to 2:
\[ \left[2(2^2) + 2\right] - \left[2(1^2) + 1\right] = (8 + 2) - (2 + 1) = 10 - 3 = 7 \]

Calculating the second integral:
\[ \int e^x \, dx = e^x \]
Evaluating from 1 to 2:
\[ e^2 - e^1 = e^2 - e \]

### Step 6: Combine the results
Now, we combine the results:
\[ A = 7 - (e^2 - e) = 7 - e^2 + e \]

### Final Result
Thus, the area of the region bounded by the curves is:
\[ A = 7 - e^2 + e \]